Lecture-19

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

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

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