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Random Variable and Its Probability Distribution

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Probability Distribution:

An arrangement of all possible values of a random variable along with
their respective probabilities is called a ‘probability distribution’ or
a ‘probability function’.
Probability distribution can be further bifurcated into:

Discrete Probability Distribution, and
Continuous Probability Distribution.

Discrete Probability Distribution: Let a discrete random variable X
assume values x1, x2, x3, ……….., xn with respective probabilities P(x1),
P(x2), P(x3), …………, P(xn). Since the random variable takes a discrete
set of values, it is also called a discrete probability distribution. A
discrete probability distribution may take the form of a table, a graph
or a mathematical equation.

A probability distribution is similar to a relative frequency
distribution with probabilities replacing relative frequencies.

A discrete probability distribution must possess the following two
properties:
                           0   d" P ( x i )    d" 1
  "P ( x i )   =   1 ,   w h i c h   m e a n s   t h a t                   t h e   s u m
 o f   p r o b a b i l i t i e s   i s   e q u a l   t o                   o n e .

 E x a m     p   l e :
 A   c o     i   n   i s   t o s s e d   t h r e e   t i m e s .     F i n d
 t h e       p   r o b a b i l i t y   d i s t r i b u t i o n   o f   t h e
 r a n d     o   m   v a r i a b l e   n u m b e r   o f   h e a d s .

 S o l u t i o n :
 S   =   { H H H ,         H H T ,     H T H ,   T H H ,         H T T ,     T H T ,
 T T H ,   T T T }

No. of Heads
X Probability of X
P(X) 0 1/8 1 3/8 2 3/8 3 1/8 Total 1
Example:
Determine whether the function   EMBED Equation.3                   for X = 1, 2, 3
and 4 can be a probability distribution.

Solution:

X P(X) 1 2/14 2 3/14 3 4/14 4 5/14 Total 1
Continuous Probability Distribution: As we known that a random variable
which can assume all possible values within a given interval is called a
continuous random variable. Within a given interval, there are an
infinite number of values. For example, t h e r e      m a y   b e    a n
 i n f i n i t e   n u m b e r   o f   w e i g h t s    b e t w e e n
 6 9 . 5   k g s   a n d   7 0 . 5   k g s .     I n    c a s e    o f    a
 c o n t i n u o u s   r a n d o m   v a r i a b l e ,
 t h e r e f o r e ,   w e   c o m p u t e   p r o b a b i l i t i e s
 f o r   v a r i o u s   i n t e r v a l s   o f   c o n t i n u o u s
 r a n d o m   v a r i a b l e ,   s u c h   a s   P ( a     d" X    d"
 b )   o r   P ( X   e" c ) .

 T h e   p r o b a b i l i t y distribution of a continuous random
variable cannot be presented in tabular form. It can be represented by
means of a formula or through a graph. The formula is necessarily in the
form of a function of the numerical values of the continuous random
variable X. For e.g., a continuous random variable can assume values
between X = 2 and X = 4 and the function is given by:

  EMBED Equation.3

The continuous probability distribution is further discussed in detail
later.

Mean   and   Variance of   a   Random Var i a b l e :
 I n     a     p r o b a   b   i l i t y   d i s t r i   b   u   t i o n   o f   a
 r a   n d   o m   v a r   i   a b l e   X ,   t h e     m   e   a n ,   a l s o
 r e   f e   r r e d   t   o     a s     M a t h e m a   t   i   c a l
 E x   p e   c t a t i o   n       o r     E x p e c t   e   d     V a l u e   ,
 a n   d     t h e   v a   r   i a n c e   a r e   d e   f   i   n e d   a s :
 ¼   =   E ( X )          =   £   X   ‡ P ( X )
 a n d   Ã 2   =          V ( X )   =   £  X 2   ‡              P ( X )
 [ E ( X ) ] 2

 D i s t r i b u t i o n   F u n c t i o n :
A function showing probabilities that a random variable X has a value
less than or equal to x is called the ‘cumulative distribution function’
or ‘distribution function of x’.

Symbolically, the cumulative distribution function, denoted by f(x) is
defined as:

     EMBED Equation.3

The cumulative distribution function has the following properties:

f(–


        -")      =   0   a n d   f ( -")   =     1 ,   w h i c h   m e a n s
 t   h a t     f ( x )    i s   a n   i n c r e a s i n g     f u n c t i o n
 r   a n g i   n g    f r o m   0   t o   1 .
 I   f   a     <   b    t h e n   f ( a )    <    f ( b )   f o r   a n y
 r   e a l     n u m b e r s    a   a n d    b .

 F   o   r   a     d   i s c r   e t e   r   a   n   d o m   v   a r i a b   l e ,
 d   i   s t r i   b   u t i o   n   f u n   c   t   i o n   i   s   o b t   a i n e d   b y
 c   u   m u l a   t   i n g     p r o b a   b   i   l i t i e   s   j u s   t   a s   w e
 o   b   t a i n   e   d   c u   m u l a t   i   v   e   d i s   t r i b u   t i o n .

 T h e   d i s t r i b u t i o n   f u n c t i o n   f o r   t h e
 p r o b a b i l i t y   d i s t r i b u t i o n   o f   t h e
 p r e v i o u s   t w o   e x a m p l e s   i s   a s   b e l o w :

 x       f ( x )     x   <   0   0    0   d" x    <   1   1 / 8                      1       d"
 x       <   2   4 / 8     2   d" x   <   3   7 / 8     x   e" 3                     1
 x       f ( x )     x   <   1   0    1   d" x    <   2   2 / 1 4                        2        d"
 x       <   3   5 / 1 4     3   d" x   <   4   9 / 1 4     x   e"
 4       1
 E   x   a m p l e   :
 C   a   l c u l a   t e   t h e   m e a n   a n d   v a r i a n c e        f o r
 t   h   e   f o l   l o w i n g   p r o b a b i l i t y
 d   i   s t r i b   u t i o n :

 X   0    1   2  3   4        5   6    7      P ( X )    0 . 1 1     0 . 2 3         0 .
3 4    0 . 1 6  0 . 1 0        0 . 0 6     0 . 0 4    0 . 0 1
 S o l u t i o n :

 X    P ( X )    X ¼ P ( X )     X   2 ¼ P ( X )        0    0 . 1 1    0    0            1
   0 . 2 3    0 . 2 3    0 . 2 3         2    0 . 3 4     0 . 6 8    1 . 3 6          3
 0 . 1 6    0 . 4 8     1 . 4 4         4    0 . 1 0    0 . 4 0    1 . 6 0           5
0 . 0 6    0 . 3 0     1 . 5 0        6     0 . 0 4    0 . 2 4    1 . 4 4        7        0
 . 0 1    0 . 0 7    0 . 4 9         T o t a l     1    2 . 4    8 . 0 6
 ¼    =   E ( X )    =    £    X     ‡ P ( X )      =     2 . 4
 Ã 2    =   V ( X )     =   £    X   2     ‡ P ( X )          [ E ( X ) ] 2         =
 8 . 0 6         ( 2 . 4 ) 2     =      2 . 3

 B i n o m i a l        P r o b a b i l i t y      D i s t r i b u t i o n :

 B i n o m i a l probability is a mathematical formula to determine
probabilities of the discrete values of a random variable called
‘Binomial Random Variable’.
The following are the conditions of Binomial Probability:

If an experiment contains only two possible outcomes, i.e., success or
failure.
The probability of ‘success’ is denoted by ‘p’ and the probability of
‘failure’ is denoted by ‘q’ where q = 1 – p or p + q = 1.
Such an experiment is repeated n times independently. In independent
repetitions, the probability p remains constant.

The number of success in n experiments is the Binomial Random Variable
and is denoted by X. The possible values of X are 0, 1, 2, 3, 4, ….., n.
The probabilities of the values of X are calculated by the following
formula:

     EMBED Equation.3

Where x = 1, 2, 3, 4, ………, n

The above formula is ‘Binomial Probability Distribution’. The two
constant quantities p and n are called the parameters of a Binomial
Distribution. The quantity q is not a separate parameter because q = 1 –
p.

Mean and Variance of a Binomial Distribution:
The mean and variance of a binomial distribution are directly evaluated
in terms of its parameters p and n.

     EMBED Equation.3

Example:
A coin is tossed 3 times. ‘Number of heads’ in 3 tosses is the random
variable X. Calculate probabilities of all possible values of X. Also
calculate mean and variance.

Solution:
Experiment: A coin is tossed for 3 times.
Success: Head

p = P(success) = P(head) = ½
n = number of times the coin is tossed = 3
x = 0, 1, 2, 3.

Now applying the Binomial Formula:
  EMBED Equation.3

P(x=0)   =    EMBED   Equation.3          =     EMBED     Equation.3
P(x=1)   =    EMBED   Equation.3          =     EMBED     Equation.3
P(x=2)   =    EMBED   Equation.3          =     EMBED     Equation.3
P(x=3)   =    EMBED   Equation.3          =     EMBED     Equation.3

Mean and Variance:

X P(X) X.P(X) X2.P(X)        0 0.125 0 0 1 0.375 0.375 0.375              2 0.375 0.75 1.5
  3 0.125 0.375 1.125        Total 1 1.5 3
  EMBED Equation.3

Hyper Geometric Probability Distribution:

It is a formula to determine the probabilities of the values for a random
variable called ‘Hyper Geometric Random Variable’.
Following are the conditions of hyper geometric random variable:

There are N items of which K are of first kind and the remaining (N – K)
are of second kind,
A sample of n items is randomly drawn without replacement from the N
items.

Number of items of first kind in the sample is the random variable X:

Possible values of X are 0, 1, 2, ………, k when
n                                            e" K                     a n d
  0 ,  1 ,   2 ,   & & & . . ,   n   w h e n   n                  <      K

 T h e       p r o b a b i l i t i e s          o f   t h e s e   v a l u e s
 a r e       c a l c u l a t e d   b y          t h e   f o r m u l a :

     E M B E D        E q u a t i o n . 3

 W h e r e      x     =     0 ,    1 ,    2 ,       3 ,     & & . . ,     k   w h e n   n
 e" k
 A n d   x      =     0 ,    1 ,    2 ,       3 ,    & & & ,      n     w h e n   n <   k

 T h e   a b o v e          f o r m u l a   i s   c a l l e d    H y p e r
 G e o m e t r i c          P r obability Distribution’. A schematic
explanation of this formula may be given as:




Example:
A committee of 3 persons is to be formed from among 3 men and 2 women.
If the selection of the committee members is random, construct the
probability distribution of the random variable ‘Number of women in the
committee’.

Solution:




Where x = 0, 1, 2, 3, ……, k; when n

                                                         e"   k
   A n d    x    =     0 ,   1 ,   2 ,     3 ,    & & . . ,    n ;   w h e n    n
 <   k

     E M B E D       E q u a t i o n . 3

     E M B E D       E q u a t i o n . 3

     E M B E D       E q u a t i o n . 3

     E M B E D       E q u a t i o n . 3

 T h e   H y p e r   G e o m e t r i c   P r o b a b i l i t y
 D i s t r i b u t i o n   o f   R V     N o .   o f   W o m e n          i n
 t h e   C o m m i t t e e     i s   a s   f o l l o w s :

 X   P ( X )     0   0 .1 1 0.6 2 0.3            Total 1
Poisson Probability Distribution:
A random variable created by counting the number of items or events in a
unit of either time or space is called a ‘Poisson Random Variable’.
Examples of Poisson random variable are the number of accidents per day
on a highway, number of cars arriving at petrol pump in a five minute
period of time, number of typing mistakes per page and number of defects
in a painted surface, etc.
A Poisson probability distribution formula assigns probabilities to
 t h e   v a l u e s   o f   t h e     P o i s s o n   R a n d o m
 V a r i a b l e   :

         E M B E D        E q u a t i o n . 3

 W h e r e           x    =     0 ,       1 ,   2 ,         3 ,     & & . .

 W   h e r e   »   ( l         a    m b d a )   i s   t h e   o n l y
 p   a r a m e t e r           o    f   t h e   d i s t r i b u t i o n                 a n d     e
 i   s   t h e   m a t         h    e m a t i c a l   c o n s t a n t
 2   . 7 1 8 2 8 & & &         .    . :

 T   h   e     n u   m b e    r     o f     e v e   n   t   s   p   e   r   u n   i t   o f
 t   i   m   e   o   r   s    p a   c e     r e m   a   i   n s     s   t a b l   e   f o r   a
 l   o   n   g   p   e r i    o d     o   f   t i   m   e   .       T   h i s     i s   t h e
 p   a   r   a m e   t e r      o   f     t h e     d   i   s t r   i   b u t i   o n
 d   e   n   o t e   d   b    y     » .
 T   h   e     n u   m b e    r     o f     e v e n t s   i n   o n e   t i m e
 p   e   r   i o d     i s      i   n d   e p e n d e n t   o f   t h e   n u m b e r
 o   f       e v e   n t s      i   n     a n o t h e r   t i m e   p e r i o d .

 E x a m p l e :
 I n   a n   i n d u s t r y ,   t h e   a v e r a g e   n u m b e r
 o f   d a m a g ed output units per week is 10. What is the probability
that there will be (i) no damaged unit in the next week, (ii) 5 damaged
units in the next week, and (iii) 15 damaged units in the next week.

Solution:
no damaged unit in the week:

X = number of damaged output units next week = 0



     »   =           a v e r a g e   n u m b e r                  o f    d a m a g e d
 u n i t s           p e r   w e e k   =   1 0

         E M B E D        E q u a t i o n . 3

 5       d a m a g e d          u n i t s       i n         t h e       n e x t    w e e k :

 X   =   n u         m   b e r   o    f   d a m a g e d             o u t p u t       u n i t s
 n e x t   w         e   e k   =      5
 »   =   a v         e   r a g e      n u m b e r   o f             d a m a g e d       u n i t s
 p e r   w e         e   k   =   1    0

         E M B E D        E q u a t i o n . 3
 1 5         d a m a g e d           u n i t s         i n       t h e           n e x t       w e e k :

 X   =   n u       m   b e r   o         f   d a m a g e d               o u t p u t           u n i t s
 n e x t   w       e   e k   =           1 5
 »   =   a v       e   r a g e           n u m b e r   o f               d a m a g e d           u n i t s
 p e r   w e       e   k   =   1         0

         E M B E D         E q u a t i o n . 3

 P   o   i s s o   n     A p p       r   o x i m   a t i o n             t o      B i n o m i a l
 D   i   s t r i   b   u t i o       n   :
 T   h   e   c o   m   p u t a       t   i o n s       i n v     o   v e
                                                                     l           d   i n   t h e
 b   i   n o m i   a   l   d i       s   t r i b   u   t i o     n   s b         e c o m e   q u     i t e
 t   e   d i o u   s     w h e       n     n   i   s     l a     r   e .
                                                                     g               I n   s u c     h
 c   a   s e s     t   h e   b       i   n o m i   a   l   d     i   t r
                                                                     s           i b u t i o n       c a n
 b   e     a p p   r   o x i m       a   t e d     t   o   a         o i
                                                                     P           s s o n
 d   i   s t r i   b   u t i o       n     w i t   h     »       =   n           ¼   p   u n d e     r
 t   h   e   f o   l   l o w i       n   g   c o   n   d i t     i o n s         :

 n       i s   v e r y   l a r g e ,
 p       i s   v e r y   s m a l l ,                   a n d
 n       ¼   p   i s   f i n i t e .

 A       f   r e q u   e   n t   l   y     u s e   d       r u l     e       o   f   t   h u m b     i s
 t   h   a   t   t h   e     a   p   p r   o x i   m a     t i o     n       i   s
 a   p   p   r o p r   i   a t   e     w   h e n     p       d"      0   .   0   5   a   n d     n   e"
 2   0   .       H o   w   e v   e   r ,     t h   e       P o i     s   s   o   n
 d   i   s   t r i b   u   t i   o   n     s o m   e t     i m e     s       p   r o v   i d e s
 c   l   o   s e   a   p   p r   o   x i   m a t   i o     n s       e   v   e   n   i   n   c a s e s
 w   h   e   r e   n       i s       n o   t   l   a r     g e       n   o   r     p     i s   v e r y
 s   m   a   l l .

 E x a m p l e :
 I n   a   v i l l a g e ,   t h e   l o c a l   g o v e r n m e n t
 a p p r o x i m a t e d   t h a t   2% of the population are infected
with seasonal flu due to absence of proper medication. What is the
probability that the number of infected persons in a random sample of 50
will be 4?

Solution:
Using binomial distribution with:

n = 50, p = 0.02 and x = 4
  EMBED Equation.3

Using Poisson approximation to the binomial with:



                                              »        =     n       ¼       p    =      5 0     ×   0 . 0 2
 =       1

         E M B E D         E q u a t i o n . 3
 T h e        P o i s s o n   p r o b a b i l i t y   i s            c l o s e     t o
 t h e        b i n o m i a l   p r o b a b i l i t y .

 M   e   a   n     a   n   d   V a r i a n c e   o   f   P o i s s o n
 D   i   s   t r   i   b   u t i o n :
 T   h   e     m   e   a   n   o f   a   P o i s s   o n   R a n d o m
 V   a   r   i a   b   l   e   i s   t h e   p a r   a m e t e r   o f    t h e
 P   o   i   s s   o   n     d i s t r i b u t i o   n   » ,   t h a t    i s :

 E ( X )           =       »

 T h e        v a r i a n c e          i s   a l s o     t h e    p a r a m e t e r
 » :

 V ( X )           =       »

 T h u s   m e a n   a n d   v a r i a n c e                o f    P o i s s o n
 d i s t r i b u t i o n   a r e   e q u a l                t o    » .

 C o n t i n u o u s   P r o b a b i l i t y                D i s t r i b u t i o n
 ( I n   D e t a i l ) :

 T h e   c o n c e p t   o f   p r o b a b i l i t y   f o r
 c o n t i n u o u s   r a n d o m   v a r i a b l e   i s
 s o m e w h a t   d i f f e r e n t   w i t h that of a discrete random
variable.
The function or the formula of continuous probability distribution is
generated and its curve is drawn on a graph paper such that:

the function is non-negative for all possible values of the random
variable, and
the total area under the curve of the function is one.

This function is called ‘probability density function’ and its curve a
‘probability curve’.

The probability of an interval from a to b is defined as the area under
the probability curve between the two vertical lines erected on the x-
axis at the points a and b.
The probability of an individual value under the continuous probability
distribution is considered zero.
Probabilities of continuous random variable are represented by areas
under the probability curve.

Normal Probability Distribution:

The most important and widely used probability density function is the
‘Normal Distribution’ where probability curve is a bell shaped
symmetrical curve:
The most mathematical form of Normal Probability Densi t y
 F u n c t i o n   i s :

         E M B E D    E q u a t i o n . 3

 W h e r e            -"       d"   x    d"    -"

 A       n o r   m a l     p   r o b a   b    i l i t y   d i   s t r   i   b   u   t   i   o n   o r
 i   t   s   p   r o b a   b   i l i t   y      c u r v e   c   h a r   a   c   t   e   r   i s e d
 b   y     t w   o   q u   a   n t i t   i    e s   ¼   a n d     Ã     c   a   l   l   e   d   t h e
 p   a   r a m   e t e r   s     o f     t    h e   d i s t r   i b u   t   i   o   n   .
 T   w   o   n   o r m a   l     c u r   v    e s   w i t h     d i f   f   e   r   e   n   t
 m   e   a n s     ¼   a   n   d   e q   u    a l   s t a n d   a r d
 d   e   v i a   t i o n   s     Ã   a   r    e   a s   b e l   o w :




 T h e   n o r m a l   c u r v e s   w i t h   d i f f e r e n t
 s t a n d a r d   d e v i a t i o n s   Ã   a n d   e q u a l
 m e a n s   ¼ :




 T w o   n o r m a l   c u r v e s   w i t h   d i f f e r e n t
 m e a n s   ¼   a n d   d i f f e r e n t   s t a n d a r d
 d e v i a t i o n s   Ã :
 A r e a          u n d e r        N o r m a l      C u r v e :

 T h e area between two limits of an interval under a normal probability
curve cannot be determined analytically.
Tables of areas evaluated numerically could have been constructed but it
would be impossible for an infinite number of normal curves for all
values of



                                             ¼   a n d   Ã .
 T h i s   p r o b l e m   i s               o v e r c o m e   b y
S t a n d a r d   N o r m a l               P r o b a b i l i t y
 D i s t r i b u t i o n      w            h o s e   m e a n   i s   z e r o   ( ¼
 =   0 )   a n d   s t a n d a             r d   d e v i a t i o n   i s   o n e
 ( Ã   =   1 ) .     T h e    s            t a n d a r d   n o r m a l
 v a r i a b l e   i s   d e n             o t e d   b y     x   :

         E M B E D           E q u a t i o n . 3

 T   h   e       t a b   l e   o f   a r e a s   u n d e r   t h e
 s   t   a n     d a r   d   n o r m a l   c u r v e   i s   u s e d   t o
 f   i   n d       a r   e a   u n d e r   n o r m a l   p r o b a b i l i t y
 c   u   r v     e :
 F   o   l l     o w i   n   g   s t e p s     a r e   i   n   v o l v e d   i n
 d   e   t e     r m i   n   i n g   t h e     a r e a     o   r   p r o b a b i l i t y
 o   f     a       p a   r   t i c u l a r     i n t e r   v   a l   o f   a   n o r m a l
 d   i   s t     r i b   u   t i o n   w i t   h   ¼   a   n   d   Ã :

 D   e   t   e   r m i   n e   t h   e   z -   v   a l u e s   f o r   e a c h
 l   i   m   i   t   o   f   i n t   e r v a   l   ,
 F   r   o   m     t h   e   n o r   m a l     a   r e a   t a b l e ,
 d   e   t   e   r m i   n e   t h   e   a r   e   a   f o r   e a c h   z - v a l u e ,
 S   u   b   t   r a c   t   t h e     s m a   l   l e r   a r e a   f r o m   t h e
 l   a   r   g   e r     o n e .

 P r e c i s e l y ,   a   v a l u e   o f   r a n d o m
 v a r i a b l e     x     c a n   b e   c o n v e r t e d                     t o
 v a l u e     z     b y :

         E M B E D           E q u a t i o n . 3

 W h e r e   ¼   a n d   Ã   a r e   t h e                     m e a n   a n d
 s t a n d a r d   d e v i a t i o n   o f                     t h e   r a n d o m
 v a r i a b l e       z .

 C o n v e r s e l y ,   t h e   z - v a l u e   c a n   b e
 c o n v e r t e d   i n t o   r a n d o m   v a r i a b l e        x   b y :

 x   =   ¼     +   Ã     ‡   z

   z     i s   t h e   n u m b e r   o f    s t a n d a r d
 d e v i a t i o n s   f r o m   o r    t o   t h e   m e a n .     A l l
 i n t e r v a l s   c o n t a i n i n g    t h e   s a m e   n u m b e r
 o f   s t a n d a r d   d e v i a t i o n s    f r o m   m e a n   w i ll
contain the same area under the curve for any normal distribution.
‘Normal Area Table’ gives an idea under the curve to the left of a z-
value. For example, for z = 1.51, the Area under Normal Curve (as shown
in the Table) is 0.9345; for z = – 2.69, the Area under Normal Curve
(from the Table) is 0.0036.
Some of the rules should be remembered:

Area to the left of z = 0 is 0.5000
Area to the left of z =   EMBED Equation.3          is 0
Area to the left of z =   EMBED Equation.3          is 1.000

Example:
A norma l      r a n d o m   v a r i a b l e    x   h a s   m e a n    µ   =
 2 4   a n   d    s t a n d a r d   d e v i a t i o n    Ã   =   1 . 8 .
 D e t e r   m i n e    z   v a l u e s   f o r   x    =   1 4 ,    1 5 . 9 ,
 2 9 . 2     a n d    3 3 .     A l s o   s h o w    t h e s e   v a l u e s
 o n   n o   r m a l    c u r v e .

 S o l u t i o n :
     E M B E D   E q u a t i o n . 3

 F o r   x     =   1 4 ;          E M B E D   E q u a t i o n . 3

 F o r   x     =   1 5 . 9 ;         E M B E D   E q u a t i o n . 3


 F o r   x     =   2 9 . 2 ;         E M B E D   E q u a t i o n . 3


 F o r   x     =   3 3 ;          E M B E D   E q u a t i o n . 3
E x a m p         l e :
A   n o r         m a l   r a n d o m               v a r i a b l e   x   h a s   m e a n
¼   =   3         6   a n d   s t a n d             a r d   d e v i a t i o n   2 . 0 5 ,
d e t e r         m i n e   t h e   v a             l u e s   o f   x   f o r   z   =
3 . 3 6 ,               1 . 8 ,   0 . 9             5   a n d   2 . 7 5 .

S o l u t i o n :
x   =   ¼   +   Ã               ‡    z

F o r   z             =        3 . 3 6 ;   x  =   3 6                         +    2 . 0 5          ×         (
3 . 3 6 )             =    2 9 . 1 1 2   H" 2 9 . 1 1

F o r   z   =       1 . 8 ;                     x       =       3 6       +       2 . 0 5       ×        (
1 . 8 )   =   3 2 . 3 1

F o r   z   =              0 . 9 5 ;  x             =       3 6       +       2 . 0 5       ×       0 . 9 5       =
3 7 . 9 4 7 5              H" 3 7 . 9 5

F o r   z   =              2 . 7 5 ;  x             =       3 6       +       2 . 0 5       ×       2 . 7 5       =
4 1 . 6 3 7 5              H" 4 1 . 6 4

E   x   a   m p   l   e   :
T   h   e     m   e   a   n   a n d   S D   o f   a   n o r m a l   r a n d o m
v   a   r   i a   b   l   e   a r e   3 4 . 5   a n d   5 . 8
r   e   s   p e   c   t   i v e l y .     F i n d   t h e   f o l l o w i n g
a   r   e   a s   :

t o   t h e   l e f t   o f   1 9 . 5
t o   t h e   r i g h t   o f   4 0
b e t w e e n   1 9 . 5   a n d   4 0

S o l u t i o n :
t o   t h e   l e f t                 o f       1 9 . 5 ,             i . e . ,            P ( x         d"
1 9 . 5 ) :

 P (     -" d" x    d" 1 9 . 5 )                            =     P (             -"   d"   z       d"
2 . 5 9 )   =  0 . 0 0 4 8

W h e r e                  E M B E D        E q u a t i o n . 3




t o   t h e               r i g h t       o f       4 0 ,         i . e . ,            P    ( x          e"
4 0 ) :
 P ( 4 0   d"   x    d"   -")    =      P ( 0 . 9 5   d"   z   d"    -")   =
 0 . 8 2 8 9

 W h e r e          E M B E D    E q u a t i o n . 3




 b e t w e e n  1 9 . 5         a n d    4 0 ,   i . e . ,     P ( 1 9 . 5      d"
 x   d" 4 0 ) :

 P ( 1 9 . 5   d" x    d" 4 0 )    =   P (     2 . 5 9          d"     z   d"
 0 . 9 5 )   =   0 . 8 2 8 9 – 0.0048 = 0.8241




Continuity Correction:

A population with unknown mean and standard deviation can be assumed a
normal population of the frequency distribution of a sample is
symmetrical. The sample mean and sample standard deviation are used as
estimates of population mean and population standard deviation
respectively.
Observations or data are always discrete, recorded up to a certain degree
of accuracy irrespective of whether the variable itself is discrete or
continuous.
When the symmetrical distribution of any data is assumed to be normal, a
continuity correction is applied to the observed values to make the data
continuous.
If the data are recorded in whole numbers, data values are considered as
mid-points of the intervals x ± 0.5, if the data are recorded up to one
decimal place, data values are considered as mid points of the intervals
x ± 0.05 and so on. It should be cleared that the 0.5 and 0.05 should be
subtracted from lower limit and added to upper limit or at most limit.

Normal   Ap p r o x i m a t i o n    t o  B i n o m i a l
 D i s   t r i b u t i o n :
 A   B   i n o m i a l   D i s t r i b u t i o n   w i t h   l a r g e               n
 a n d     m o d e r a t e   p    c a n  b e   a p p r o x i m a t e d
 t o     a   N o r m a l   D i s t r i b u t i o n   w i t h   m e a n               ¼
 =     n ¼ p    a n d                  E M B E D       E q u a t i o n . 3                 :

 ¼     =    n ¼ p

       E M B E D      E q u a t i o n . 3

 E x a m p l e :
 A   p a i r   o f   d i c e   i s   r o l l e d                      f o r     800 times.
What is the probability that a total of 6 occur:

at least 100 times, and
between 150 to 300 times.

Solution:
n = 800
p =    EMBED Equation.3                               E = {15, 24, 33, 42, 51}
q =   EMBED Equation.3

     EMBED Equation.3

     EMBED Equation.3




                              P r o b a b i l i t y             o f       a t  l e a s t
 1 0 0   t i m e s ,           i . e . ,   P ( 1 0 0             d"   x      d" 8 0 0 )        o r
 P ( 9 9 . 5   d" x            d" 8 0 0 . 5 ) :

       E M B E D      E q u a t i o n . 3

 P (       1 . 1 9    d"      z        d"       7 0 . 4 9 )

 F r o m     N o r m a l   A r e a   T a b l e                       t h e      N o r m a l
 A r e a   c o r r e s p o n d i n g   t o                         1 . 1 9      i s
 0 . 1 1 7 0

 =     1        0 . 1 1 7 0                 =    0 . 8 8 3 0

 P r o b a b i l i t y   o f   b e t w e e n                     1 5 0  a n d   3 0 0
 t i m e s ,   i . e . ,   P ( 1 3 0   d" x                      d" 3 0 0 )   o r
 P ( 1 4 9 . 5   d" x    d" 3 0 0 . 5 ) :

       E M B E D      E q u a t i o n . 3

 P ( 1 . 8 8         d"   z       d"        1 9 . 3 7 )

 F r o m         N o r m a l   A r e a   T a b l e     t h e   N o r m a l
 A r e a       c o r r e s p o n d i n g   t o   1 . 8 8   i s   0 . 9 6 9 9

 =     1        0 . 9 6 9 9                 =    0 . 0 3 0 1

 f ( x )
 1
7 / 8

6 / 8

5 / 8

4 / 8

3 / 8

2 / 8

1 / 8

N C n

K C x

N             k C n         x

R a n d o m           S a m p l e   w i t h o u t   R e p l a c e m e n t

N

n

K

N         K

x

n     x

N o r m a l           P r o b a b i l i t y   C u r v e

¼ 1                      ¼ 2

à 1

à 2

¼ 1       <    ¼ 2
à 1       =    à 2

à 1

¼ 1       =    ¼ 2
à 1       <    à 2

¼ 1       =    ¼ 2

à 2
 Ã 2

                          ¼ 1                       ¼ 2

 ¼ 1          <       ¼ 2
 Ã 1          <       Ã 2

 Ã 1

 N        =       5

 n        =       3

 K        =       2

 N                K       =     3

 x   =
 0 ,   1 ,                2

 n        – x =       3, 2, 1

     14               15.9                     24       29.2            33

– 5.56                            – 4.5             0            2.89        5

              19.5                    34.5

              – 2.59                      0

From the ‘Normal Area Table’, 0.0048 is shown corresponding to z = – 2.59

0.8289

                  34.5                    40

                      0                   0.95

0.8241

                  19.5                34.5                  40

              – 2.59                      0                 0.95

0.0048

As shown in ‘Normal Area Table’

S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}


              3               2               2         1           2        1   1   0
S = {HHH, HHT, HTH, HTT, THH, THT, TTH, TTT}


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   ÿÿÿÿÿÿÿÿÿÿÿÿ                                                     C o m p
O b j                                                                ÿÿÿÿ
                                      f           O b j I n f o
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   DS Equation
   Equation.3 ô9²q
                       }Ó<   ìmI èyI

ƒP ‚( ƒX ‚) †=
ƒx †+ ˆ1       ˆ1 ˆ4
                                                       þÿ
ÿÿÿÿ Î         À        F     Microsoft Equation 3.0
   DS EqE q u a t i o n   N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                          X              _ 1 2 1 6
0 6 7 6 8 2                                     "   •
    Î      À      F     Ò•Îê·Æ Ò•Îê·Æ   O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                          C o m p
O b j
   ÿÿÿÿ                f    uation
   Equation.3 ô9²q
                           xËT       8•I `ƒI

ƒf ‚( ƒx ‚) †=
ƒx †+ ˆ1 ˆ8     —        —
     ˆ2 †d"ƒx †d"ˆ4
                                         þÿ
ÿÿÿÿ Î      À      F       Microsoft Equation 3.0
 DS Equation
 Equation.3 ô9²q
                                               O b j I n f o
                                       ÿÿÿÿ
  ÿÿÿÿ                                                 E q u a t i o n
N a t i v e
  ÿÿÿÿÿÿÿÿÿÿÿÿ
  p       _ 1 2 1 5 8 9 0 2 6 5
                  Î      À      F   Ò•Îê·Æ    Ò•Îê·Æ              O l e

 ÿÿÿÿÿÿÿÿÿÿÿÿ
              C o m p O b j
               ÿÿÿÿ
   f          O b j I n f o
       ÿÿÿÿ    ÿÿÿÿ                                                 E q u a
 t i o n    N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                          T        _ 1 2 1 5
8 9 1 9 4 5                                                  ÿÿÿÿÿÿÿÿ     Î
     À       F      <¨Îê·Æ <¨Îê·Æ                  }Ó8          ìmI èyI

ƒf ‚( ƒx ‚) †= ƒP ‚( ƒx †d"ƒx ‚)

              þÿ
  ÿÿÿÿ Î      À      F    Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                                                     }Óp        ìmI èyI

ƒP ‚( ƒx ‚) †=   ,
ƒn
ƒC
ƒx
†Å"ƒp
ƒx
†Å"ƒq
ƒn   O l e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                                  C o m p
O b j                                                             ÿÿÿÿ
                                     f          O b j I n f o
                                         ÿÿÿÿ    ÿÿÿÿ
                            E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ                                        Œ        †"ƒx

                                                þÿ
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   DS Equation
   Equation.3 ô9²q
                                                     }Óˆ        ìmI èyI

ƒE ‚( ƒX ‚) †= ƒn †Å"ƒp   ƒV ‚( ƒX ‚) †= ƒn †Å"ƒp †Å
"‚( ˆ1 †"ƒp ‚)    —
       †= ƒn †Å"ƒp †Å"ƒq                                   _ 1 2 1
5 8 9 2 1 6 1
   6       Î       À        F       s—Îê·Æ   s—
Îê·Æ               O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                                               C o m p
O b j                                                                          ÿÿÿÿ
                                         f            O b j I n f o
                                             ÿÿÿÿ      ÿÿÿÿ
                                E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ                                                 ¤           _ 1 2 1 6
0 4 1 3 5 2                                                            ¤   ÿÿÿÿ-
    Î      À       F       <¨Îê·Æ   <¨Îê·Æ                     O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ
             C o m p O b j
      -
   ÿÿÿÿ                                           !   f
                                                          þÿ
  ÿÿÿÿ Î       À       F    Microsoft Equation 3.0
DS Equation
Equation.3 ô9²q
ÐØ       ìmI èyI
  ,
ˆ3
ƒC
ˆ0
†Å"
  ˆ1   ˆ2   –(   –)
ˆ0
†Å"
  ˆ1 ˆ2       –( –)
ˆ3 †"ˆ0
†= ˆ1 †× ˆ1 †× ˆ0 ‚. ˆ1 ˆ2 ˆ5 †= ˆ0 ‚. ˆ1 ˆ2 ˆ5
           O b j I n f o
   ÿÿÿÿ!    ÿÿÿÿ                                   #            E q u a t
i o n    N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                    $   ô        _ 1 2 1 6
0 4 1 5 4 7                                               J   ,    $    Î
     À       F     <¨Îê·Æ <¨Îê·Æ               O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                    (
                                                        þÿ
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DS Equation
Equation.3 ô9²q
ÐÜ       dµI Ü‘I
  ,
ˆ3
ƒC
ˆ1
†Å"
  ˆ1   ˆ2   –(   –)
ˆ1
†Å"
  ˆ1 ˆ2      –( –)
ˆ3 †"ˆ1
†= ˆ3 †× ˆ0 ‚. ˆ5 †× ˆ0 ‚. ˆ2 ˆ5 †= ˆ0 ‚. ˆ3 ˆ7 ˆ5
          C o m p O b j
   #   %   ÿÿÿÿ                                    )   f          O b j I
n f o                                                     ÿÿÿÿ&    ÿÿÿÿ
                                 +           E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ                                    ,   ø        _ 1 2 1 6
0 4 1 6 4 1                                               ÿÿÿÿÿÿÿÿ)     Î
     À       F    <¨Îê·Æ <¨Îê·Æ                O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                    0                 C o m p
O b j                                                        (   *    ÿÿÿÿ
                                 1   f          O b j I n f o
                                        ÿÿÿÿ+    ÿÿÿÿ
               3           E q u a t i o n
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   ÿÿÿÿÿÿÿÿÿÿÿÿ                                    4   ø
                                                           þÿ
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DS Equation
Equation.3 ô9²q
ÐÜ       zI „‘I
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ˆ3
ƒC
ˆ2
†Å"
  ˆ1   ˆ2   –(   –)
ˆ2
†Å"
  ˆ1 ˆ2       –( –)
ˆ3 †"ˆ2
†= ˆ3 †× ˆ0 ‚. ˆ2 ˆ5 †× ˆ0 ‚. ˆ5 †= ˆ0 ‚. ˆ3 ˆ7 ˆ5    •
         _ 1 2 1 6 0 4 1 8 7 3
   '   1    .    Î      À      F    <¨Îê·Æ <¨Îê·Æ                O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                   8              C o m p
O b j                                                    -
   /   ÿÿÿÿ                                   9   f           O b j I n f
o                                                     ÿÿÿÿ0    ÿÿÿÿ
                             ;
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DS Equation
Equation.3 ô9²q
ÐØ       °®I ì I
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ˆ3
ƒC
ˆ3
†Å"
  ˆ1   ˆ2   –(   –)
ˆ3
†Å"
  ˆ1 ˆ2      –( –)
ˆ3 †"ˆ3
†= ˆ1 †× ˆ0 ‚. ˆ1 ˆ2 ˆ5 †× ˆ1 †= ˆ0 ‚. ˆ1 ˆ2 ˆ5
        E q u a t i o n   N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                    <   ô        _ 1 2 1 6
0 4 2 3 2 3                                               ÿÿÿÿÿÿÿÿ3     Î
     À      F     <¨Îê·Æ Àc±Îê·Æ               O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                    @              C o m p
O b j                                                     2   4    ÿÿÿÿ
                                 A   f
                                      þÿ
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DS Equation
Equation.3 ô9²q
Ðø       8•I `ƒI
ƒE ‚( ƒX ‚) †= ƒn †Å"ƒp †= ˆ3 †×
  ˆ1 ˆ2    †= ˆ1 ‚. ˆ5 ƒV ‚( ƒX ‚) †= ƒn †Å"ƒp †Å"‚(
ˆ1 †"ƒp ‚)    —    †= ƒn †Å"ƒp †Å"ƒq †= ˆ3 †×
ˆ1   ˆ2   †×
  ˆ1     ˆ2†= O b j I n f o
             ÿÿÿÿ5    ÿÿÿÿ                                 C
E q u a t i o n    N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                     D        _ 1 2 1 5
9 5 5 0 1 0                                             ÿÿÿÿÿÿÿÿ8    Î
     À      F    Àc±Îê·Æ Àc±Îê·Æ              O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                     I          ˆ0 ‚.
ˆ7 ˆ5
                                               þÿ
  ÿÿÿÿ Î          À   F   Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                       MÕœ   ìmI èyI

ƒP ‚( ƒx ‚) †=
     ,
ƒk
ƒC
ƒx
†Å",
ƒN †"ƒk
ƒC
ƒn †"ƒx
  ,
ƒN
ƒC
ƒn                     C o m p O b j
                  7   9   ÿÿÿÿ                                            J   f
       O b j I n f o
ÿÿÿÿ:   ÿÿÿÿ                                      L                 E q u a t i o
 n   N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                       M   ¸          _ 1 2 1 6
1 4 8 2 0 9                                                     ÿÿÿÿÿÿÿÿ=    Î
     À       F   Àc±Îê·Æ Àc±Îê·Æ                O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                       P                   C o m p
O b j                                                           <     >    ÿÿÿÿ
                                Q   f           O b j I n f o
                                        ÿÿÿÿ?    ÿÿÿÿ
               S           E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                       T   à
                                                              þÿ
  ÿÿÿÿ Î        À    F   Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                               ;ÐÄ   ¸€I à‚I

ƒP ‚( ƒx †= ˆ0 ‚) †=
     ,
ˆ2
ƒC
ˆ0
†Å",
ˆ3
ƒC
ˆ3
  ,
ˆ5
ƒC
ˆ3
†=
  ˆ1 †× ˆ1 ˆ1 ˆ0    †= ˆ0 ‚. ˆ1
     _ 1 2 1 6 1 4 8 3 4 9
;   E   B    Î      À      F    Àc±Îê·Æ Àc±Îê·Æ                  O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                     X                C o m p
O b j                                                      A     C    ÿÿÿÿ
                                 Y   f           O b j I n f o
                                         ÿÿÿÿD    ÿÿÿÿ
              [
                    þÿ
  ÿÿÿÿ Î      À      F    Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                               ;ÐÄ   ü¨I È©I

ƒP ‚( ƒx †= ˆ1 ‚) †=
     ,
ˆ2
ƒC
ˆ1
†Å",
ˆ3
ƒC
ˆ2
  ,
ˆ5
ƒC
ˆ3
†=
  ˆ2 †× ˆ3 ˆ1 ˆ0    †= ˆ0 ‚. ˆ6
     E q u a t i o n   N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                   \   à        _ 1 2 1 6
1 4 8 4 0 6                                               ÿÿÿÿÿÿÿÿG    Î
     À      F    Àc±Îê·Æ Àc±Îê·Æ              O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                   `               C o m p
O b j                                                     F   H    ÿÿÿÿ
                                 a  f
                                     þÿ
  ÿÿÿÿ Î      À      F   Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                               ;ÐÄ   °}I \„I

ƒP ‚( ƒx †= ˆ2 ‚) †=
     ,
ˆ2
ƒC
ˆ2
†Å",
ˆ3
ƒC
ˆ1
  ,
ˆ5
ƒC
ˆ3
†=
   ˆ1 †× ˆ3 ˆ1 ˆ0    †= ˆ0 ‚. ˆ3
        O b j I n f o
ÿÿÿÿI    ÿÿÿÿ                                      c                E q u a t i o
 n    N a t i v e
    ÿÿÿÿÿÿÿÿÿÿÿÿ                                       d   à              _ 1 2 1 5
9 5 6 2 6 5                                                           m      L    Î
      À       F   Àc±Îê·Æ Àc±Îê·Æ              O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                        h
                                                               þÿ
  ÿÿÿÿ Î      À      F    Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                       MÕ\   ìmI èyI

ƒP ‚( ƒx ‚) †=
  ƒe
†"„»
†Å"„»
ƒx
ƒx ‚!
                 þÿ
  ÿÿÿÿ Î     À        F   Microsoft Equation 3.0
     DS
Eq    C o m p O b j                                                                K
 M     ÿÿÿÿ                                      i    f                 O b j I n f o
                                                         ÿÿÿÿN           ÿÿÿÿ
                            k               E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                           l    x              _ 1 2 1 6
1 4 9 0 9 0                                                         @     ^      Q    Î
     À       F    à ¹Îê·Æ à ¹Îê·Æ                    O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                           n                     C o m p
O b j                                                               P     R      ÿÿÿÿ
                                    o   f            O b j I n f o
                                         ÿÿÿÿS        ÿÿÿÿ
               q            E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                           r    ð              uation
   Equation.3 ô9²q
                                                              ;ÐÔ             ¸€I à‚I

ƒP ‚( ƒx †= ˆ0 ‚) †=
  ƒe
†"„»
†Å"„»
ƒx
ƒx ‚!   †=
  ƒe
†"ˆ1 ˆ0
†× ˆ1 ˆ0
ˆ0
ˆ0 ‚!   †= ˆ0 ‚. ˆ0 ˆ0 ˆ0 ˆ0 ˆ4 ˆ5 ˆ4
                                                     þÿ
  ÿÿÿÿ Î      À      F    Microsoft Equation 3.0
   DS
Eq_ 1 2 1 6 2 3 1 2 1 3                                                           ÿÿÿ
ÿÿÿÿÿV    Î      À      F      à ¹Îê·Æ à ¹Îê·Æ                        O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                         v                     C o m p
O b j                                                             U     W      ÿÿÿÿ
                                     w   f           O b j I n f o
                                             ÿÿÿÿX    ÿÿÿÿ
               y            uation
   Equation.3 ô9²q
                                                            zÈÈ               I Ô I

ƒP ‚( ƒx †= ˆ5 ‚) †=
  ƒe
†"„»
†Å"„»
ƒx
ƒx ‚!   †=
  ƒe
†"ˆ1 ˆ0
†× ˆ1 ˆ0
ˆ5
ˆ5 ‚!   †= ˆ0 ‚. ˆ0 ˆ3 ˆ7 ˆ8
                                                  þÿ
  ÿÿÿÿ Î      À      F   Microsoft Equation 3.0
   DS EqE q u a t i o n   N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                       z   ä        _ 1 2 1 6
1 4 9 5 0 0                                                   ÿÿÿÿÿÿÿÿ[    Î
     À      F    à ¹Îê·Æ à ¹Îê·Æ                  O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                       ~                C o m p
O b j                                                         Z    \    ÿÿÿÿ
                                   •    f       þÿÿÿþÿÿÿƒ     „   …    þÿÿÿþÿÿÿ
ˆ   þÿÿÿþÿÿÿ‹   Œ    •   Ž    þÿÿÿþÿÿÿ‘    þÿÿÿþÿÿÿ”    •    –
   þÿÿÿþÿÿÿ™   þÿÿÿþÿÿÿœ     •   þÿÿÿþÿÿÿ     þÿÿÿþÿÿÿ£    þÿÿÿþÿÿÿ¦    þÿÿÿþÿ
ÿÿ©   þÿÿÿþÿÿÿ¬    þÿÿÿþÿÿÿ¯    þÿÿÿþÿÿÿ²    þÿÿÿþÿÿÿµ    þÿÿÿþÿÿÿ¸    þÿÿÿþÿÿ
ÿ»   þÿÿÿþÿÿÿ¾   þÿÿÿþÿÿÿÁ     Â   þÿÿÿþÿÿÿÅ    þÿÿÿþÿÿÿÈ     É   þÿÿÿþÿÿÿÌ
þÿÿÿþÿÿÿÏ    Ð  þÿÿÿþÿÿÿÓ     þÿÿÿþÿÿÿÖ    þÿÿÿþÿÿÿÙ    þÿÿÿþÿÿÿÜ    þÿÿÿþÿÿÿß
   þÿÿÿþÿÿÿâ   þÿÿÿþÿÿÿå     þÿÿÿþÿÿÿè    þÿÿÿþÿÿÿë   þÿÿÿþÿÿÿþÿÿÿþÿÿÿð     þÿ
ÿÿþÿÿÿþÿÿÿþÿÿÿõ    þÿÿÿþÿÿÿø    þÿÿÿþÿÿÿû    þÿÿÿþÿÿÿþ    ÿ        uation
   Equation.3 ô9²q
                                                         ;ÐÐ        À€I è‚I

ƒP ‚( ƒx †= ˆ5 ‚) †=
  ƒe
†"„»
†Å"„»
ƒx
ƒx ‚!   †=
  ƒe
†"ˆ1 ˆ0
†× ˆ1 ˆ0
ˆ1 ˆ5
ˆ1 ˆ5 ‚!    †= ˆ0 ‚. ˆ0 ˆ3 ˆ4 ˆ7
                                                    þÿ
  ÿÿÿÿ Î        À     F    Microsoft Equation 3.0
   DS
Eq O b j I n f o                                                             ÿÿÿ
ÿ]   ÿÿÿÿ                                      •                E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                       ‚     ì            _ 1 2 1 6
1 5 0 2 9 0                                                      Y   r      `    Î
     À       F  à ¹Îê·Æ à ¹Îê·Æ                    O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                       †                   C o m p
O b j                                                            _   a     ÿÿÿÿ
                                    ‡   f          O b j I n f o
                                           ÿÿÿÿb    ÿÿÿÿ
               ‰              E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                       Š     $            _ 1 2 1 6
2 3 1 1 8 2                                                      ¸   T      e    Î
     À       F   à ¹Îê·Æ   ,ÂÎê·Æ              uation
   Equation.3 ô9²q
                                                           ;Ð            À€I è‚I

ƒP ‚( ƒx ‚) †= ,
ƒn
ƒC
ƒx
†Å"ƒp
ƒx
†Å"ƒq
ƒn †"ƒx
†= ,
ˆ5 ˆ0
ƒC
ˆ4
†× ˆ0 ‚. ˆ0 ˆ2
ˆ4
†× ˆ0 ‚. ˆ9 ˆ8
ˆ5 ˆ0 †"ˆ4
†= ˆ0 ‚. ˆ0 ˆ1 ˆ4 ˆ5
                                                            O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                       •                   C o m p
O b j                                                            d   f     ÿÿÿÿ
                                    •   f          O b j I n f o
                                           ÿÿÿÿg    ÿÿÿÿ
               ’              E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                       “     Ü             þÿ
  ÿÿÿÿ Î      À      F     Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                               zÈÀ   À€I è‚I

ƒP ‚( ƒx †= ˆ4 ‚) †=
  ƒe
†"„»
†Å"„»
ƒx
ƒx ‚!   †=
  ƒe
†"ˆ1
†× ˆ1
ˆ4
ˆ4 ‚!    †= ˆ0 ‚. ˆ0 ˆ1 ˆ5 ˆ3
                                                                            _ 1
2 1 5 9 5 7 6 0 7                                                      ÿÿÿÿÿÿÿÿj
    Î      À      F       ,ÂÎê·Æ   ,ÂÎê·Æ                O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                     —
             C o m p O b j
      i   k   ÿÿÿÿ                                        ˜      f           O b j
 I n f o                                                             ÿÿÿÿl    ÿÿÿÿ
                                     š              þÿ
  ÿÿÿÿ Î       À      F    Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                       MÕ   ìmI èyI

ƒf ‚( ƒx ‚) †=
  ˆ1
  ˆ2 „À

†Å"„à   †Å"ƒe
†"
ˆ1   ˆ2
ƒx †"„¼ „Ã   –(   –)
ˆ2
           þÿ
  ÿÿÿÿ Î        À   F   Microsoft Equation 3.0
   DS EqE q u a t i o n    N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                   ›    ¼            _ 1 2 1 5
9 7 7 8 1 6                                                 h   |      o    Î
     À      F     ,ÂÎê·Æ   ,ÂÎê·Æ             O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                   ž                  C o m p
O b j                                                       n   p     ÿÿÿÿ
                                    Ÿ    f   uation
   Equation.3 ô9²q
                                                      •Ó,           ìmI èyI

ƒz †=
ƒx †"„¼ „Ã

 þÿ
ÿÿÿÿ Î     À          F   Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                                                        O b j I n f o
                                                ÿÿÿÿq    ÿÿÿÿ
                   ¡               E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                          ¢     H        _ 1 2 1 6
1 5 1 7 0 4                                                        ÿÿÿÿÿÿÿÿt    Î
     À       F         ,ÂÎê·Æ   ,ÂÎê·Æ                O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                          ¤                 C o m p
O b j                                                              s   u    ÿÿÿÿ
                                         ¥   f        O b j I n f o
                                              ÿÿÿÿv    ÿÿÿÿ
               §                 E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                          ¨     H        _ 1 2 1 5
9 7 8 3 5 3                                                        ÿÿÿÿÿÿÿÿy    Î
     À       F         ,ÂÎê·Æ   ,ÂÎê·Æ                   ;Ð,          À€I è‚I

ƒz †=
ƒx †"„¼ „Ã

 þÿ
ÿÿÿÿ Î     À          F   Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                                                             •Ó0           ìmI èyI

†"ˆ3 ‚. ˆ5    †"ˆ4 ‚. ˆ0
                         O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                         ª                   C o m p
O b j                                                              x   z     ÿÿÿÿ
                                   «      f          O b j I n f o
                                          ÿÿÿÿ{       ÿÿÿÿ
               -             E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                         ®     L            _ 1 2 1 5
9 7 8 4 0 0                                                        w          ~    Î
     À       F   ÀÍÉÎê·Æ ÀÍÉÎê·Æ                     O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                         °                   C o m p
O b j                                                              }   •     ÿÿÿÿ
                                   ±      f          O b j I n f o
                                           ÿÿÿÿ€    ÿÿÿÿ
              ³
                      þÿ
  ÿÿÿÿ Î      À        F   Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                                                    •Ó(   ¥I ,¥I

ˆ3 ‚. ˆ5    ˆ4 ‚. ˆ0

              þÿ
  ÿÿÿÿ Î      À        F   Microsoft Equation 3.0
   DS EqE q u a t i o n   N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                  ´    D              _ 1 2 1 6
1 5 1 7 3 2                                                 O    •      ƒ    Î
     À      F    ÀÍÉÎê·Æ ÀÍÉÎê·Æ             O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                  ¶                    C o m p
O b j                                                      ‚   „     ÿÿÿÿ
                                 ·      f   uation
   Equation.3 ô9²q
                                                     ;Ð,             À€I è‚I

ƒz †=
ƒx †"„¼ „Ã

 þÿ
ÿÿÿÿ Î     À          F   Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                                                   O b j I n f o
                                         ÿÿÿÿ…    ÿÿÿÿ
                  ¹           E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                     º     H          _ 1 2 1 6
1 5 1 7 6 5                                                   ÿÿÿÿÿÿÿÿˆ    Î
     À       F    ÀÍÉÎê·Æ ÀÍÉÎê·Æ                O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                     ¼                    C o m p
O b j                                                     ‡        ‰    ÿÿÿÿ
                                    ½ f          O b j I n f o
                                       ÿÿÿÿŠ    ÿÿÿÿ
               ¿            E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                     À     Œ            _ 1 2 1 6
1 5 1 8 3 4                                                   †    •    •    Î
     À       F   ÀÍÉÎê·Æ ÀÍÉÎê·Æ                    ;Ðp             è•I 0¥I

ƒz †=
ƒx †"„¼ „Ã   †=
ˆ1 ˆ4 †"ˆ2 ˆ4   ˆ1 ‚. ˆ8   †= †"ˆ5 ‚. ˆ5 ˆ6

                                    þÿ
ÿÿÿÿ Î     À      F     Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                     ;Ðt   ˆ­I •ƒI

ƒz †=
  ƒx †"„¼ „   O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                       Ã                    C o m p
O b j                                                       Œ    Ž        ÿÿÿÿ
                                   Ä   f           O b j I n f o
                                           ÿÿÿÿ•    ÿÿÿÿ
              Æ               E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                       Ç   •            Ã    †=
ˆ1 ˆ5 ‚. ˆ9 †"ˆ2 ˆ4   ˆ1 ‚. ˆ8   †= †"ˆ4 ‚. ˆ5

                                   þÿ
ÿÿÿÿ Î     À       F   Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                     ;Ðt   LŸI tŸI

ƒz †=
ƒx †"„¼ „Ã   †=
     ˆ2 ˆ9 ‚. ˆ2 †"ˆ2 ˆ4       ˆ1 ‚. ˆ8     †= ˆ2_ 1 2 1 6 1 5 1 8 9
 2                                                       ÿÿÿÿÿÿÿÿ’    Î      À
     F        ÀÍÉÎê·Æ ÀÍÉÎê·Æ                 O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                              Ê                   C o m p
O b j                                                             ‘       “     ÿÿÿÿ
                                          Ë   f          O b j I n f o
                                               ÿÿÿÿ”    ÿÿÿÿ
               Í                    E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                              Î    •             _ 1 2 1 6
1 5 1 9 4 1                                                         ‹     ®      —
    Î      À           F        ÀÍÉÎê·Æ€õÒÎê·Æ                 O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                              Ñ                   C o m p
O b j                                                                 –
   ˜   ÿÿÿÿ                                              Ò   f            ‚. ˆ8 ˆ9

                                                   þÿ
     ÿÿÿÿ Î        À       F      Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                     ;Ð`   T‹I ˆ“I

ƒz †=
ƒx †"„¼ „Ã   †=
  ˆ3 ˆ3 †"ˆ2 ˆ4   ˆ1 ‚. ˆ8   †= ˆ5
                                                  O b j I n f o
                                       ÿÿÿÿ™     ÿÿÿÿ
               Ô            E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                      Õ    |          _ 1 2 1 6
1 5 4 0 2 1                                                   ÿÿÿÿÿÿÿÿœ    Î
     À       F   €õÒÎê·Æ€õÒÎê·Æ                 O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                      ×                   C o m p
O b j                                                      ›      •     ÿÿÿÿ
                                    Ø f           O b j I n f o
                                       ÿÿÿÿž     ÿÿÿÿ
                Ú           E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                      Û    €             _ 1 2 1 6
1 5 4 2 3 7                                                   š   ©      ¡    Î
     À       F    €õÒÎê·Æ€õÒÎê·Æ              þÿ
  ÿÿÿÿ Î       À      F    Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                     ;Ðd   À€I è‚I

ƒz †=
  ˆ1 ˆ9 ‚. ˆ5 †"ˆ3 ˆ4 ‚. ˆ5 ˆ5 ‚. ˆ8   †= †"ˆ2 ‚. ˆ
5 ˆ9
   þÿ
  ÿÿÿÿ Î      À      F    Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q                                   O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                              Ý                C o m p
O b j                                                     ¢    ÿÿÿÿ
                          Þ   f           O b j I n f o
                                  ÿÿÿÿ£    ÿÿÿÿ
               à     E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                              á   t
                                                      ;ÐX         À€I
 è‚I
ƒz †=
ˆ4 ˆ0 †"ˆ3 ˆ4 ‚. ˆ5   ˆ5 ‚. ˆ8   †= ˆ0 ‚. ˆ9 ˆ5

 þÿ
ÿÿÿÿ Î     À       F   Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                                                          •Ó(             ìmI èyI

„Æ=
  ƒn ƒp ƒq
  _ 1 2 1 5 9 7 8 8 9 0                                                         ÿÿÿ
ÿÿÿÿÿ¦    Î      À      F      €õÒÎê·Æ€_ëÎê·Æ                     O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                      ã                      C o m p
O b j                                                           ¥     §      ÿÿÿÿ
                                    ä   f         O b j I n f o
                                          ÿÿÿÿ¨    ÿÿÿÿ
               æ             E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                      ç     D         _ 1 2 1 6
2 2 8 5 0 6                                                     ÿÿÿÿÿÿÿÿ«    Î
     À       F   €_ëÎê·Æ€_ëÎê·Æ                 O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                      é                      C o m p
O b j                                                           ª     ¬      ÿÿÿÿ
                                    ê   f

                               þÿ
  ÿÿÿÿ Î      À      F      Microsoft Equation 3.0
DS Equation
Equation.3 ô9²q
                  zÈ
   tmI pyI
ˆ5     ˆ3 ˆ6
                    þÿ
ÿÿÿÿ Î         À     F   Microsoft Equation 3.0
   DS
Eq O b j I n f o                                                           ÿÿÿ
ÿ-   ÿÿÿÿ                                   ì                 E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                    í     <             _ 1 2 1 6
2 2 8 5 6 8                                                    Ÿ   c      °    Î
     À       F  €_ëÎê·Æ€_ëÎê·Æ                O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                    î                    C o m p
O b j                                                          ¯   ±     ÿÿÿÿ
                                    ï   f       O b j I n f o
                                        ÿÿÿÿ²    ÿÿÿÿ
               ñ           E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                    ò     @          _ 1 2 1 6
2 2 9 6 0 0                                                    ÿÿÿÿÿÿÿÿµ    Î
     À       F   €_ëÎê·Æ€_ëÎê·Æ           uation
   Equation.3 ô9²q
                                                        zÈ$            ¬¤I ÜnI
ˆ3 ˆ1   ˆ3 ˆ6        P
                      þÿ
ÿÿÿÿ Î       À         F   Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                                                      zÈ`             I Ô I

„¼†= ƒn †× ƒp †= ˆ8   O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                   ó                   C o m p
O b j                                                       ´   ¶      ÿÿÿÿ
                                ô      f       O b j I n f o
                                       ÿÿÿÿ·    ÿÿÿÿ
               ö          E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                   ÷    |            ˆ0 ˆ0 †×
ˆ5     ˆ3 ˆ6   †= ˆ1 ˆ1 ˆ1 ‚. ˆ1   q
                                             þÿ
ÿÿÿÿ Î         À      F    Microsoft Equation 3.0
   DS Equation
   Equation.3 ô9²q
                         zÈ´   À€I è‚I

„Æ=
  ƒn †× ƒp †× ƒq

†=
  ˆ8 ˆ0 ˆ0 †×
ˆ5   ˆ3 ˆ6   †×
  ˆ3 ˆ1   ˆ3 ˆ6

†=
  ˆ9 ˆ5 ‚. ˆ7

†= ˆ9 _ 1 2 1 6 2 2 9 6 1 4
    ³   ½  º    Î      À      F         @‡ôÎê·Æ@‡ôÎê·Æ                   O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                        ù                  C o m p
O b j                                                            ¹   »     ÿÿÿÿ
                                    ú    f         O b j I n f o
                                        ÿÿÿÿ¼       ÿÿÿÿ
               ü           E q u a t i o n
 N a t i v e
   ÿÿÿÿÿÿÿÿÿÿÿÿ                                        ý    Ð            _ 1 2 1 6
2 2 9 7 1 7                                                      ÿÿÿÿÂ      ¿    Î
     À       F   @‡ôÎê·Æ@‡ôÎê·Æ                  O l e

   ÿÿÿÿÿÿÿÿÿÿÿÿ                                                          C o m p
O b j                                                            ¾   À    ÿÿÿÿ
                                         f       þÿÿÿþÿÿÿ       þÿÿÿþÿÿÿ
        þÿÿÿþÿÿÿ
   þÿÿÿþÿÿÿ                þÿÿÿ                            þÿÿÿ        -

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DOCUMENT INFO
Description: bank reconcilation statement,circular flow of income, closing enteries, coefficient of correlation, concepts of National income, Consumption Functions, Determination of National income,Financial statement, Flow of Funds accounts, Index numbers, Introduction to statistics,investment, macro economics, measures of central tendency, measure of depersion, measure of shape distribution, merchandise inventory, micro macro economics, Micro Economics, multiplier and accelerator, probability, public finance, random variable,regression,sample distribution theory, statistics dynamics and comparative, theofy of employment, trade cycle