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Unformatted text preview: Discrete random variables and probability distributions Chapter 3 (3.4  3.6) 1 The Binomial Probability Distribution 2 The Binomial Probability Distribution Consider the following random experiments and random variables: Flip a coin ten times. Let X = number of heads obtained. A worn machine tool produces 1% defective parts. Let X=number of defective parts in the next 25 produced. In the next 20 births at a hospital, let X = the number of female births. 3 The Binomial Probability Distribution The random variable in each case is a count of the number of trials that meet a specified criterion. Each trial can be resulting in either a ‘success’ or ‘failure’. 4 5 Binomial Experiment An experiment for which the following four conditions are satisfied is called a binomial experiment . 1. The experiment consists of a sequence of n trials, where n is fixed in advance of the experiment. 2. The trials are identical, and each trial can result in one of the same two possible outcomes, which are denoted by success ( S ) or failure ( F ). 3. The trials are independent. 4. The probability of success is constant from trial to trial: denoted by p . Given a binomial experiment consisting of n trials, the binomial random variable X associated with this experiment is defined as X = the number of S ’s among n trials. Binomial Experiment 6 The true Binomial experiment requires an infinite or conceptual population or replacement when sampling from a finite population. Suppose each trial of an experiment can result in S or F , but the sampling is without replacement from a finite population of size N…then the experiment is not a true Binomial . However, if the sample size n is at most 5% of the population size, the experiment can be analyzed as though it were exactly a binomial experiment. 7 Binomial Experiment Example: Sam Young is a 71.6% free throw shooter. What is the likelihood that he will make 7 of his next 10 free throws? 8 Probability mass function (pmf) of a Binomial random variable (rv) Because the pmf of a binomial rv X depends on the two parameters n and p , we denote the pmf by b ( x ; n , p ) x n x n x p p C p n x b = ) 1 ( ) , ; ( , x = 0,1,2…n otherwise 9 Example: A card is drawn from a standard 52 card deck. If drawing a club is considered a success, find the probability of a. exactly one success in 4 draws (with replacement). b. no successes in 5 draws (with replacement)....
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This note was uploaded on 04/17/2011 for the course ENGR 0020 taught by Professor Rajgopal during the Spring '08 term at Pittsburgh.
 Spring '08
 Rajgopal

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