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					Mc100405351
Aqib Farooq

Answer No .1
  a) The probability density function of a random variable is given as
  F(x) = 2(x-1)        1<x<2
           0          Elsewhere
  Calculate second moment about origin=?
  (As we know the moment about origin)
   u1 = E (x) ……… first moment
   u2 = E (x) ………. second moment
   u3 = E (x)…………third moment
   u4 = E (x)…………fourth moment

    As per rave
    P (z≥-1.1)
    -1.1 se bare

  In the second diagram 0.5+0.5=1
  Shaded area is raga

  In the third diagram
  P (z≥ -1.1) =0.5+0.3643
  =0.8643 Answer

 Second Moment about Origin

 Second moment about origin = E (x²)

E (x²) = -∞ ∫∞ x2. f(x) dx

E (x²) = 1∫2 x2. 2 (x+1) dx

E (x²) = 21∫2 x2. (x-1) dx

E (x²) = 21∫2 (x3 – x2) dx

E (x²) = 2 [x4/5 – x3/3]2

E (x²) = 2 [(2)4/4 – (2)3/3 - ( (1)4/4 – (1)3/3 ) ]

E (x²) = 2 (48-32/12) - (3-9/12)

E (x²) = 2 [16/12 + 1/12]
 E (x²) = 2 (17/12)


E (x²) = 17/6 Answer

Part b)

F (x,y) = x (1+3y2) / 4

= 0 , elsewhere

  0<x<2,0<y<1
Find the marginal probability density function of x =?

g(x) = -∞ ∫∞ f (x,y) dy

0<y<1

 = 0 ∫1 x (1+ 3y2) /4 .dy

 = 0 ∫1 x (1+ 3y2) /4 .dy

 = 1/4 0 ∫1 (x + 3xy2) dy
                       Y=1
 = 1/4 [ xy+ 3xy3 /3]10

 = 1/4 [xy +xy3]1

 =1/4 [x(1) + x(1)3 – (x(0) + x(0)3]

 =1/4 [x+x-0]

 =1/4 (2x)

G(x) = x/2      for (0 < x < 2)

Answer No.2
   a) what is the probability that a poker hand of 5 cards contain exactly 2 Aces ?

   n (S) = 52C5

   Let a donate exactly 2 aces
        (4C2)   (48C3)
P(A) = ______________
         (52C5)

P(A)= 6 (17296)
      2598960


      103776
P(A)= ______          Answer
      2598960

Part b)
A pair of dice is rolled 180 times. By using the normal approximation to binominal
distribution , find the probability of a total of 7 occurs at least 25 times ?

Probability of 7 is

(1,6) (2,5) (3,4) (4,3) (5,2) (6,1)

P = 6/36 = 1/36

Now

n=180
p= 1/6
q= 1-p = 1-1/6 = 5/6

µ = n p =180*1/6 =30

♪=√npq

♪ = √ 180*1/6*5/6

♪=5

Now

Z=X-µ
   ♪

Z= X-30
    5
At least 25 times
Which means
P (x ≥ 25)
Continuity correction

              Working Notes
when we go to continuous form discrete we add or less 0.5

In this case he said at least 25 times which means 25 is include that is why we less 0.5
from 25 for continuity correction.

P ( x ≥ 24.5)
Now

Z=X-µ
     ♪
Z = X-30
     5

Z = 24.5 – 30
        5
Z = -1.1

Now
Go to page # 225 of Sta301 handouts and see the value of 1.1 in 0 column.

Because it is 1.10 last pay zero ha agar 1.12 rate ho to 1.1 ki value column 2 me
dekhein.
Now 1.1 in column zero = 0.3643
Is ka matlab ha k origin say -1.1 ka fasla 0.3643 ha.

				
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