Tutorial 6 by gegeshandong

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									Tutorial 6

Mode and Median of Random
Variable, Gamma Distribution



                               1
Mode of random variable
 Discrete  R.V.:
  If T = k is the mode, then,
  P(T  k )  P(T  i) for i
 Continuous   R.V.:
  If a is the mode, then
   f(a)  f(b) for b
 Position   of the maximum in p.d.f / p.m.f
                                               2
Median of random variable
 Discrete  R.V.:
  If T = i is the median, then,
   i                         i 1

   P (T  k )  1
  k 1
                     2
                         and  P (T  k )  1
                             k 1
                                                2
                                                    for i  1

               P (T  i )  1                       for i  1
                                    2
 Continuous   R.V.:
                                    a           1
  If a is the median, then F (a)   f ( x)dx 
                                                        2
                                                                3
Example - Binomial R.V

X   is a random variable with parameter (n,p),
  find the median and mode of X?
 Median: a minimum k such that
            k
                          1
            P( X  r )  2
           r 1
            k
                                    1
          nCr (1  p)   nr
                                p 
                                 r

           r 1                     2

                                              4
Example - Binomial R.V

 Mode:   a maximum k such that
                 P{ X  k}
                              1
                P{ X  k  1}
         Ck (1  p ) n  k p k
          n
                      n  k 1 k 1
                                    1
      nCk  1(1  p )         p
                 (n  k  1) p
                                    1
                   k (1  p )
                                  k  (n  1) p
                                                  5
Example – Exponential R.V

X   is a random variable with p.d.f,
           e  x if 0  x  
  f ( x)  
           0       otherwise
  where is a positive constant.

 Find its mode and median.
 Find its generating function, and thereby its
  skewness and kurtosis.
                                                  6
Example – Exponential R.V

 Mode:
  As f(x) is strictly decreasing function, its
  maximum occurs at x=0.
   0 is the mode.
               a           1
 Median:
             0 f ( x)dx  2
               a           1       ln 2
             0 e dx  2  a  
                    x


                                                 7
Example – Exponential R.V
   Moment Generating Function:
                                                 
    M (t )  E[e ]   e e
               tX             tx    x
                                          dx 
                         0                        t
                    E[( X   ) 3 ]
   Skewness 
                    3
               E[( X   ) 4 ]
    Kurtosis 
                    4
   By finding E[X], E[X²], E[X³],…from M(t),
    skewness and kurtosis can be found.
                                                        8
Gamma Distribution
   A random variable is said to have a gamma
    distribution with parameters (t, ), if its density
    function is given by
                  e  x (x)t 1
                                   x0
        f ( x)        (t )
                 0                 x0
                 
        Where (t ) is the gamma function,
                     
        define by  e  y y t 1dy
                    0                                     9
Gamma Distribution
                  
 Since (t )   y t 1e  y dy,
                 0

                                         
 then (t )  e y    y   t 1
                                        e  y (t  1) y t  2 dy
                                  0      0

                             
               (t  1)  e  y y t 2 dy
                            0

               (t  1)(t  1)


   (n)  (n  1)! for positive n
                                                                     10
Gamma Distribution

 The   cumulative density function,
   F ( x)  P{ X  x}
                           t 1
                                  x    j
            1 e    x
                           
                           j 0     j!
                                              if x  0


                                                         11
Mean of Gamma Distribution
 ConsiderX as a gamma random variable
 with parameter t and .
              1 
             (t ) 0
    E[ X ]           xe x (x) t 1 dx

               1      

             (t ) 0
                       e x (x) t dx

               1      

             (t ) 0
                       x x (x) t d (x)

                                              12
Mean of Gamma Distribution
                1     

             (t ) 0
    E[ X ]             e  y y t dy

             (t  1)
           
              (t )
                 t!
           
              (t  1)!
             t
           
              
 Mean   of Gamma(t , ) = t/ 
                                       13
Variance of Gamma Distribution
 First   calculate E[X2]:
             1  2 x
            (t ) 0
   E[ X ] 
          2
                     x e (x) t 1 dx

              1       

             (t ) 0
           2           e x (x)t 1dx

              1       

             (t ) 0
           2           e x (x) t 1d (x)

              1       

             (t ) 0
           2           e  y t t 1d ( y )
                                                14
Variance of Gamma Distribution
             (t  2)
   E[ X ]  2
        2

              (t )
            t (t  1)
                2
                     
  Var( X )  E[ X 2 ]  ( E[ X ])2
                t (t  1)      t
                            ( )2
                     2       
                 t
            
                2
Variance       of Gamma(t , ) = t/ ²   15
Mode of Gamma Distribution
 Since
                          e  x (x) t 1
               f ( x)                              if x  0
                               (t )
                                                    x         t 1  x
  f ' ( x)            [ (t  1)( x)   t 2
                                                e            (x) e        ]
               (t )
      
            x  0  )x ( ' f
   )1  t(

 Mode    of Gamma(t, ) = (t-1)/
                                                                                 16
HW3 Q10 Hints.
 Given   the mean and mode.
 i) First find the value of t and ’s.
 ii) Once you find the t, , you can plot the
  income distributions.
  P{ X  x}  1  G ( x;  , n)
                        n 1
                            (x) k
            1  e x            for 0  x  
                       k 0   k!
  income distribution : Gamma(n, )
                                                   17
Income Distribution
                 x
           e      ( x )   t 1
f ( x)                            if x  0
                  (t )

>> x=[0:60000];
>> t=4;
>> m=1/3000;
>> y=m*exp(-1*m*x).*(m*x).^(t-1)/gamma(t);
>> plot(y)
                                              18
Income Distribution




                      19
Gini Index

     1    

     
G            G ( x)[1  G ( x)]dx
         0




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