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LectureNotes5 - Discrete Random Variables and Their...

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Unformatted text preview: Discrete Random Variables and Their Probability Distribution: Part II Cyr Emile M’LAN, Ph.D. [email protected] Discrete Random Variable: Part II – p. 1/28 Introduction ♠ Text Reference : Introduction to Probability and Its Application, Chapter 4. ♠ Reading Assignment : Sections 4.3-4.6, February 25-March 2 In this set of notes, we will discuss several common but useful distribution models for discrete random variables. Discrete Random Variable: Part II – p. 2/28 Bernoulli Distribution Many experiments (public opinion poll, consumer preference, drug testing, quality control in industrial setting) have only two outcomes: yes/no, head/tail, male/female, fail/pass, success/failure, defective/non-defective. Bernoulli Trial (i) Two possible outcomes: success or failure . (ii) p = P ( success ) and q = P ( failure ) = 1- p . Let X = braceleftBigg 1 , if success , , if failure . Then, X has a Bernoulli distribution . Discrete Random Variable: Part II – p. 3/28 Bernoulli Distribution The probability mass function is given by x 1 p ( x ) 1- p p or simply, p ( x ) = p x q 1- x , y = 0 , 1 , The mean and variance are E [ X ] = p, and V ( X ) = E [ X 2 ]- ( E [ X ]) 2 = p- p 2 = p (1- p ) = pq, Note that X 2 = X for this case. A simple example of a Bernoulli trial is to toss a coin once and to let X = 1 if the head faces up and X = 0 if the tail comes up. Discrete Random Variable: Part II – p. 4/28 Binomial Distribution Definition 3.5 : A binomial experiment possesses the following proper- ties: 1. Each experiment consists of a fixed number, n , of inde- pendent and identical Bernoulli trials. 2. The random variable of interest is X , the num- ber of successes observed during the n trials. For an experiment to be a binomial experiment the characteristics above must all be satisfied. probability mass function X has a binomial distribution if p ( x ) = parenleftbigg n x parenrightbigg p x q n- x , x = 0 , 1 , 2 ,... ,n, where < p < 1 . Discrete Random Variable: Part II – p. 5/28 Binomial Distribution We use the following notation to denote this distribution: X ∼ Binomial ( n,p ) . Another View of Binomial Random Variable Let X i = braceleftBigg 1 if “success" in the i th Bernoulli trial , if “failure" in the i th Bernoulli trial . Then X 1 , X 2 , ... , X n are independent and identically distributed (i.i.d.) as the Bernoulli distribution with p being the probability of success. Now, write X = n summationdisplay i =1 X i . Then, X ∼ Binomial ( n,p ) . Discrete Random Variable: Part II – p. 6/28 The Binomial Distribution 0 2 4 6 8 1 0 1 2 0.00 0.05 0.10 0.15 0.20 0.25 0.30 probability distribution B in o m ia l ( n = 1 2 , p = .8 5 ) 2 4 6 8 10 12 0.0 0.1 0.2 0.3 probability distribution B in o m ial ( n = 12, p = .08) 0 2 4 6 8 1 0 1 2 0.00 0.05 0.10 0.15 0.20 0.25 0.30 probability distribution B in o m ia l ( n = 1 2 , p = .1 5 ) 0 2 4 6 8 1 0 1 2 0.00 0.05 0.10 0.15 0.20 probability distribution B in o m ia l ( n = 1 2 , p = .5 0 ) Discrete Random Variable: Part II – p. 7/28 The Binomial Distribution...
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LectureNotes5 - Discrete Random Variables and Their...

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