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# PROBABILITY STAT430

Penn

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This 17 page Class Notes was uploaded by Orval Funk on Monday September 28, 2015. The Class Notes belongs to STAT430 at University of Pennsylvania taught by Staff in Fall. Since its upload, it has received 15 views. For similar materials see /class/215433/stat430-university-of-pennsylvania in Statistics at University of Pennsylvania.

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Date Created: 09/28/15

STAT 430510 Probability Hui Nie Lecture 9 June 91h 2009 Review 0 Discrete Random Variables 0 Expected Value and Variance o Binomial Random Variable o Poisson Random Variable Geometric Random Variable o X is said to be a geometric random variable with parameter p it its pml is given by PXn1ipquot p n12 o X represent the number of trials until getting one success Each trial is independent with success probability p Example 0 Xnumber of tosses of a fair coin until getting a head The pmt of X is PX n 05 1 x 05 o X is a geometric random variable with parameter 05 Example 0 An urn contains N white and M black balls Balls are randomly selected one at a time until a black ball is obtained If we assume that each selected ball is replaced before the next one is drawn what is the probability that a Exactly n draws are needed b At least k draws are needed Example Solution 0 Let Xnumber of draws needed Then X is a geometric random variable with p MM N Thus M MN 7 1 n1 MN MN MNquot PX 2 k 2 PX n 7M1 NW1 nk Mean and Variance If X is a geometric random variable with parameter p o EX l9 0 EX2 2ng o VarX 1 Negative Binomial Random Variable o X is called a negative binomial random variable with parameters r p it its pml is given by PX n p391ipquotnnr7r17 o X represent the number of trials until a total of r successes is accumulated Again each trial is independent with success probability p Mean and Variance If X is a negative binomial rv with parameters r p o EX lg 2 7 1 o EXii 71 o VarX Hypergeometric Random Variable o X is called a hypergeometric random variable with parameter n N m if its pml is given by Paow io1n I o X represent the number of white balls that you get if you draw 17 out of N balls of which m are white balls Mean and Variance If X is a hypergeometric rv with parameters 17 N m o Em 0 EX21 quotquot i 1 o Varm quot NE 1 7 Expected Value of Sums of Random Variables 0 Proposition For random variables X1 X2 Xn HEX1 Z EXi i1 i1 Example 0 Find the expected value of the sum obtained when n fair dice are rolled 0 Let Xbethe sum 0 let Y be the upturned valueon die i EX HZ Yi 271 EY 35quot Example o Toss a fair coin until at least one head and one tail has been seen Let N be the number of times a selection is made Calculate EN Method of Indicators 0 For any event A the indicator of A IA equals to 1 if A occurs and 0 otherwise 0 The expected value of IA is the probability of A E A PM o If X is the number of events that occur among some collection of events A A then EX PA1 PAn Example 0 Suppose that we want to match 10 couples into 10 pairs What is the expected number of husbands that will be paired with their wives 0 Let X be the number of husbands that will be paired with their wives e Let I be the indicator whether ith husband is matched with his wife EXEZ21ii 215 2211 Expected Value of Functions of FLV o If EX2 1 EY2 2 EXY 1 calculate EX YX72Y EX YX 7 2Y 7 EX2 XY7 2XY7 232 7 EX2 7 EXY 7 2502 74

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