Lecture 11-2007 - Binomial Random and Normal Random...

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1 Binomial Random and Normal Random Variables: Lecture XI I. Bernoulli Random Variables A. The Bernoulli distribution characterizes the coin toss. Specifically, there are two events 0,1 X with 1 X occurring with probability p . The probability distribution function   PX can be written as:   1 [ ] 1 x x P X p p  B. Next, we need to develop the probability of XY where both X and Y are identically distributed. If the two events are independent, the probability becomes:             1 1 2 , 1 1 1 x y x y x y x y P X Y P X P Y p p p p p p   C. Now, this density function is only concerned with three outcomes   0,1,2 Z X Y . 1. There is only one way each for 0 Z or 2 Z . a) Specifically for 0 Z , 0 X and 0 Y . b) Similarly, for 2 Z , 1 X and 1 Y . 2. However, for 1 Z either 1 X and 0 Y or 0 X or 1 Y . Thus, we can derive:                   20 0 2 1 0 2 0 1 1 0 0 1 1 1 0 2 [ 0] 1 1 1, 0 0, 1 11 21 P Z p p P Z P X Y P X Y p p p p pp P Z p p      D. Next we expand the distribution to three independent Bernoulli events where   0,1,2,3 Z W X Y .
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This note was uploaded on 07/18/2011 for the course AEB 6933 taught by Professor Carriker during the Fall '09 term at University of Florida.

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Lecture 11-2007 - Binomial Random and Normal Random...

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