chapter5 - Introduction Introduction to Probability and...

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2011/09/13 1 Introduction to Probability and Statistics Thirteenth Edition Chapter 5 Several Useful Discrete Distributions Introduction • Discrete random variables take on only a finite or countably infinite number of values. • Three discrete probability distributions serve as models for a large number of practical applications: The binomial random variable The Poisson random variable The hypergeometric random variable The Binomial Random Variable • The coin-tossing experiment is a simple example of a binomial random variable. Toss a fair coin n = 3 times and record x = number of heads. x p(x) 0 1/8 1 3/8 2 3/8 3 1/8 The Binomial Random Variable • Many situations in real life resemble the coin toss, but the coin is not necessarily fair, so that P(H) 1/2. Example: A geneticist samples 10 people and counts the number who have a gene linked to Alzheimer’s disease. Person Coin: Head: Tail: Number of tosses: P(H): Has gene Doesn’t have gene n = 10 P(has gene) = proportion in the population who have the gene.
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2011/09/13 2 The Binomial Experiment 1. The experiment consists of n identical trials. 2. Each trial results in one of two outcomes , success (S) or failure (F). 3. The probability of success on a single trial is p and remains constant from trial to trial. The probability of failure is q = 1 – p. 4. The trials are independent . 5. We are interested in x , the number of successes in n trials. Binomial or Not? • Very few real life applications satisfy these requirements exactly. • Select two people from the U.S. population, and suppose that 15% of the population has the Alzheimer’s gene. • For the first person, p = P(gene) = .15 • For the second person, p P(gene) = .15, even though one person has been removed from the population. The Binomial Probability Distribution • For a binomial experiment with n trials and probability p of success on a given trial, the probability of k successes in n trials is . 1 ! 0 1 ) 2 )...( 2 )( 1 ( ! )! ( ! ! . ,... 2
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chapter5 - Introduction Introduction to Probability and...

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