Page 183 - DMTH404_STATISTICS

Unit 13: Moment Generating Function - Continue

2. Continuing the example above, the joint moment generating function of a Notes
2 × 1 standard normal random vector X is

 1 T   1 1 2 
2

M (t) exp t t  exp t  t
X    1 2 
 3   3 2 
3. Let X be a 2 × 1 discrete random vector and denote its components by X and X . Let the
1 2
support of X be
T
RX = {[1 1] , [2 2] , [0 0] }
T
T
and its joint probability mass function be
 1 T

 3 if x [1 1]

 1 if x [2 2] T

 3
p (x) = 
X 1
 if x [0 0] T

 3
 0 otherwise


Derive the joint moment generating function of X, if it exists.
4. Let
X = [X X ] T
1 2
be a 2 × 1 random vector with joint moment generating function
1 1
M , (t ,t )   exp(t  2t )
X
2
1
1
1 X
2
2
3 3
Derive the expected value of X .
1
Answers: Self Assessment
1. (c) 2. (d) 3. (a) 4. (b)
13.7 Further Readings

Books Introductory Probability and Statistical Applications by P.L. Meyer
Introduction to Mathematical Statistics by Hogg and Craig

Fundamentals of Mathematical Statistics by S.C. Gupta and V.K. Kapoor

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