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ch05t - Ch.5 1 Models for Probability Distributions...

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Ch.5 Models for Probability Distributions 1 Introduction In this chapter we develop models for the probability distributions of random variables. Each model will pertain to an entire class of random variables which share a common, up to a some parameters , probabilistic behavior. The population parameters, such as the expected value and variance of the random variable in any given experiment, can be expressed in terms of the model parameters of the assumed model. In parametric statistical inference, the model parameters are the focus of inference. The simplest random variable is one that takes only two values and is called Bernoulli random variable. The two values can always be re-coded to 0 and 1, so the sample space of a Bernoulli random variable will always be taken to be S = { 0 , 1 } . The random variables Y in Examples 2 and 3 of Chapter 3 are examples of Bernoulli random variables. The random variables X in Examples 4 and 5 of Chapter 3 are also Bernoulli when n = 1. These examples illustrate the wide variety of Bernoulli experiments, i.e. of experiments that give rise to a Bernoulli random variable. The advantage of modeling is that the probabilistic structure of any Bernoulli random variable follows from that of the prototypical one. In the prototypical Bernoulli experiment, X takes the value 1 with probability p and the value 0 with probability 1 - p . Thus, the pmf of X is x 0 1 p ( x ) 1 - p p and its cdf is x 0 1 F ( x ) 1 - p 1 We have already seen that the expected value and variance of a Bernoulli random variable X are μ X = p, σ 2 X = p (1 - p ) . 1
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The pmf and cdf of any specific Bernoulli random variable is of the above form for some value of the model parameter p . Thus, if X takes the value 0 or 1 if a randomly selected item is defective or non-defective, respectively, and the probability of a non-defective item is 0 . 9, the pmf and cdf of X follow from those of the prototypical pmf by replacing p by 0 . 9, i.e. x 0 1 p ( x ) .1 .9 With more complicated experiments, we will see that models serve not only as a convenient classification of experiments with similar probabilistic structure, but also as approxima- tions to the true probability distribution of random variables. This will be particularly the case with continuous random variables where there is often little justification for the choice of a particular model. In the following two sections we introduce the most common models for probability dis- tributions of discrete and continuous random variables, respectively. The last section introduces models for joint distributions. 2 Models for Discrete Random Variables 2.1 The Binomial Distribution Suppose that n independent Bernoulli experiments are performed, and that each exper- iment results in 1 with the same probability p . The total number, X , of experiments resulting in 1 is a binomial random variable with parameters n and p . Thus the possible values of X are 0 , 1 , · · · , n .
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