CHAPTER 8 MORE DISCRETE PROBABILITY MODELS 80 of sixes on...

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Chapter 8 More Discrete Probability Models 8.1 Introduction In last week’s lecture we started to look at discrete probability models. This week we look at two of the most common models for discrete data: the binomial distribution and the Poisson distribution. 8.2 The Binomial Distribution In many surveys and experiments data is collected in the form of counts. For example, the number of people in the survey who bought a CD in the past month, the number of people who said they would vote Labour at the next election, the number of defective items in a sample taken from a production line, and so on. All these variables have common features: 1. Each person/item has only two possible responses or “outcomes” (Yes/No, Defective/Not defective etc) — this is referred to as a trial which results in a success or failure . 2. The survey/experiment takes the form of a random sample — the responses are indepen- dent . 3. The probability of a success in each trial is p (in which case the probability of a failure is 1 - p ). 4. We are interested in the random variable X , the total number of successes out of n trials. If these conditions are met then X has a binomial distribution with index n and probability p . We write this as X Bin ( n, p ) , which reads as “ X has a binomial distribution with index n and probability p ”. Here, n and p are known as the “parameters” of the binomial distribution. Example Recall the dice rolling example from last week. We were interested in the number of sixes obtained from three rolls of a 6-sided die. Treating each roll of the die as a trial , with a six representing a success and “not a six” representing a failure , we can see that we have n = 3 independent trials, each with probability of success p = 1 / 6 . Thus if X represents the number 79
CHAPTER 8. MORE DISCRETE PROBABILITY MODELS 80 of sixes on the 3 rolls, we have that X has a binomial distribution with parameters n = 3 and p = 1 / 6 , that is X Bin (3 , 1 / 6) . 8.2.1 Probability calculations How can we work out probabilities from a binomial distribution? For example, what is P ( X = 2) in the dice rolling example? Well, as was mentioned last week, there is a formula that allows us to work out such probabilities for any values of n and p . The probability that X takes the value r , that is, that there are r successes out of n trials, can be calculated using the following formula: P ( X = r ) = (# ways to get r successes out of n trials) × P ( r successes ) × P ( n - r failures ) = n C r × p r × (1 - p ) n - r , r = 0 , 1 , . . . , n, where n C r is the number of combinations of r objects out of n
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