III. Famous Discrete Distributions: The Binomial and Poisson Distributions


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1 III. Famous Discrete Distributions: The Binomial and Poisson Distributions Up to this point, we have concerned ourselves with the general properties of categorical and continuous distributions, illustrated with somewhat arbitrary examples. However, there are particular distributions that are well understood and have wide applicability. We will first cover two specific types of categorical variables: those that follow the binomial and Poisson distributions. Both of these distributions model the probability of observing a certain number of events over a period of time or within a physical space. They are used widely in all branches of science and engineering. Later, we will spend some time discussing the most famous continuous distribution: the normal distribution or bell curve.
2 The Bernoulli Distribution As a way of understanding the binomial distribution, it helps to consider the simplest categorical random variable: a binary variable, which can take only one of two values, such as heads or tails, yes or no, male or female, dead or alive, etc. We typically code a binary random variable X as 1 or 0. We refer arbitrarily to X = 1 as a success, and X = 0 as a failure. How you choose to define success and failure is entirely up to you. If X is binary with probability of success p, then we say X ~ Bernoulli(p). That is, the pmf for X is P(X = 1) = p and P(X = 0) = 1 p. Note also that E(X) = (1)p + (0)(1 p) = p, and Var(X) = (1) 2 p + (0) 2 (1 p) p 2 = p p 2 = p(1 p).
3 The Binomial Distribution Suppose that we have n independent Bernoulli trials, each with probability of success p. The binomial distribution determines the probability of observing a given number of successes out of the n trials. Example III.A We flip a fair coin 10 times, and observe the number of tosses that result in heads. The number of heads out of 10 flips follows the binomial distribution.
4 Example III.B We survey 500 randomly selected students at USU, and ask them whether they think that President Obama is doing a good job ( yes or no ). The number who respond yes is a binomially distributed random variable. The Binomial PMF If X follows the binomial distribution, with number of independent trials given by n and probability of success given by p, we say that X ~ Binomial(n, p). The pmf of X is given by n x n x P( X = x) = p (1 p), for x = 0, K, n. x Note also that E(X) = np, and Var(X) = np(1 p).
5 Example III.C Sixteen percent of senior citizens (men and women over the age of 65) in the U.S. suffer from diabetes. Assume that the elderly in Cache County are representative of the U.S. population, and suppose that we randomly sample 25 Cache seniors. What is the probability that exactly two of them are diabetic? What is the probability that no more than two are diabetic? What is the average number of diabetic subjects you'd expect to see out of this sample of this size? What is the variance of the number of diabetics for this sample? Suppose that 8 of the 25 are diabetic. Is this evidence that the diabetes rate in Cache is higher than the national average?
6 The Poisson Distribution Like the binomial distribution, the Poisson distribution is used to model count data. The distinction between the two distributions, however, is that we use the binomial distribution to model the probability of some count out of a fixed, finite number of trials. The Poisson distribution does not depend on a fixed number of trials the range of a Poisson random variable is 0, 1, 2, The Poisson distribution is especially useful for modeling the occurrence of relatively rare events. Example III.D An engineer observes traffic flow through an intersection during the period of an hour. The number of vehicles that will pass through is a Poisson random variable.
7 Example III.E A physicist is interested in the per minute intensity of particle emissions from a radioactive substance. The number of emitted particles can be considered a Poisson random variable. The Poisson PMF There is a single parameter that determines the distribution of a Poisson random variable X: the rate parameter µ. We often refer to µ as the rate because it turns out that E(X) = µ. In other words, µ represents the average count per unit of time or space. Given µ, the Poisson pmf is x µ µ e P( X = x) =, for x = x! 0,1, 2,... Note also that Var(X) = µ. The mean and variance of a Poisson random variable are the same.
8 Example III.F According to data from the Logan Police Department, an average of 5.75 traffic accidents occur daily in Logan City. Given that the daily number of traffic accidents follows a Poisson distribution, what is the probability of 3 reported accidents on any given day? What's the probability that there are more than 3 reported accidents on a given day? What is the expected value and variance for the number of traffic accidents on a given day? Suppose that over the course of a work week (Monday through Friday), the LPD handles 20 accident reports. Is this an "unusual" week?
9 The Poisson Approximation to the Binomial Distribution It turns out the Poisson distribution can provide a good approximation of binomial probabilities. This is especially true for a binomial random variable with relatively large n and small p. Can you think of a heuristic explanation for this? Suppose that you have a random variable X ~ Binomial(n,p), and you wish to use a Poisson approximation for the pmf of X. What mean and variance would you use for this Poisson distribution? Use the Poisson approximation to compute the probabilities in Example II.C. Does the approximation work well? Why or why not?
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