Lecture-19

# Lecture-19 - Lecture 19. Generating Functions and Moment...

This preview shows pages 1–2. Sign up to view the full content.

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Lecture 19. Generating Functions and Moment Generating Functions This lecture is related to the material in Chapter 11. Recall the following from the last lecture: Definition. Let X be a random variable that takes values in the set { , 1 , 2 , . . . } and has pmf f . The generating function of X is G X ( t ) := f (0) + f (1) t + f (2) t 2 + = summationdisplay i =0 f ( i ) t i In Lecture 18, I called this p X ( t ). But we use p for so many other things that it seems like a good idea to use a different letter. G has the advantage that it is the first letter of generating. The generating function is useful because of the useful way in which it keeps track of all the information in the pmf, as the applications we see below will demonstrate. Remark. Note that if X is defined on the probability space , then G X ( t ) = summationdisplay f ( ) t X ( ) . Example. The Binomial Theorem states: ( A + B ) m = m summationdisplay i =0 parenleftbigg m i parenrightbigg A i B m i . If X is binomial ( m, p ), then f ( i ) = ( m i ) p i (1 p ) m i , so G X ( t ) = m summationdisplay i =0 f ( i ) t i = m summationdisplay i =0 parenleftbigg m i parenrightbigg ( pt ) i (1 p ) m...
View Full Document

## Lecture-19 - Lecture 19. Generating Functions and Moment...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online