slides_4_ranvecs

# 93 moment generating functions if x is an m 1 random

• Notes
• 107

This preview shows pages 93–103. Sign up to view the full content.

93

This preview has intentionally blurred sections. Sign up to view the full version.

Moment Generating Functions If X is an m 1 random vector, we can define its moment generating function from m to 0, . For m 1 vectors t , X t E exp t X  The MGF is useful only insofar it is finite for t in a neighborhood of zero. But now we need to define the neighborhood using a distance measure in m . This is usually done using Euclidean length. 94
For an m 1 vector u , its Euclidean length, denote u , is the nonnegative number u ‖ j 1 m u j 2 1/2 The Euclidean distance between two vectors is t u ‖ j 1 m t j u j 2 1/2 95

This preview has intentionally blurred sections. Sign up to view the full version.

An open neighborhood around zero with radius 0 is defined as N 0 u m : u So, for the MGF to be useful, we must assume that there exists 0 such that X t E exp t X  for all t N 0 . 96
As in the scalar case, we can use the MGF to find moments of the random vector X . In particular X t 0 t X 0 E X X 2 t t 0 t 2 X 0 E XX 97

This preview has intentionally blurred sections. Sign up to view the full version.

We can then compute Var X E XX E X E X If the MGF of X is finite in a neighborhood of zero then it is easy to see that each X j has a finite MGF on an interval about zero. Denote these 1 , 2 ,..., m . We can always write X t E exp t X  E exp j 1 m t j X j E j 1 m exp t j X j by a key property of exp  . 98
If we now assume X 1 , X 2 ,..., X m are independent then E j 1 m exp t j X j j 1 m E exp t j X j  j 1 m j t j , that is X t X t 1 , t 2 ,..., t m j 1 m j t j . 99

This preview has intentionally blurred sections. Sign up to view the full version.

There is a converse to this statement, too. Namely, if for t j j , j for j 0 we can show X t j 1 m j t j , then X 1 , X 2 ,..., X m are independent. The MGF is particularly convenient for establishing facts about the multivariate normal distribution. 100
There also is an important relationship between the MGF of a sum and the MGFs of the summands when the summands are independent. Let X 1 , X 2 ,..., X m be independent random variables with MGFs j . Define Y X 1 X 2 ... X m . Y inherits its distribution from that of the random vector X . If the X j are independent, that distribution is often much easier to characterize using the MGF of Y . This is because Y t X 1 X 2 ... X m t j 1 m j t 101

This preview has intentionally blurred sections. Sign up to view the full version.

Proof : We again use a key property of exp  along with independence: Y t E exp tY  E exp t X 1 ... X m  E j 1 m exp tX j j 1 m E exp tX j  j 1 m j t 102
This is the end of the preview. Sign up to access the rest of the document.
• Fall '12
• Jeff
• Variance, Probability theory, probability density function

{[ snackBarMessage ]}

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern