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# Distribs - CHAPTER 10 Statistical Distributions In this...

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CHAPTER 10 Statistical Distributions In this chapter, we shall present some probability distributions that play a central role in econometric theory. First, we shall present the distributions of some discrete random variables that have either a finite set of values or that take values that can be indexed by the entire set of positive integers. We shall also present the multivariate generalisations of one of these distributions. Next, we shall present the distributions of some continuous random variables that take values in intervals of the real line or over the entirety of the real line. Amongst these is the normal distribution, which is of prime importance and for which we shall consider, in detail, the multivariate extensions. Associated with the multivariate normal distribution are the so-called sam- pling distributions that are important in the theory of statistical inference. We shall consider these distributions in the final section of the chapter, where it will transpire that they are special cases of the univariate distributions described in the preceding section. Discrete Distributions Suppose that there is a population of N elements, Np of which belong to class A and N (1 p ) to class A c . When we select n elements at random from the population in n successive trials, we wish to know the probability of the event that x of them will be in A and that n x of them will be in A c . The probability will be a ff ected by the way in which the n elements are selected; and there are two ways of doing this. Either they can be put aside after they have been sampled, or else they can be restored to the population. Therefore we talk of sampling without replacement and of sampling with replacement. If we sample with replacement, then the probabilities of selecting an element from either class will the same in every trial, and the size N of the population will have no relevance. In that case, the probabilities are governed by the binomial law. If we sample without replacement, then, in each trial, the probabilities of selecting elements from either class will depend on the outcomes of the previous trials and upon the size of the population; and the probabilities of the outcomes from n successive trials will be governed by the hypergeometric law. Binomial Distribution When there is sampling with replacement, the probability is p that an ele- ment selected at random will be in class A , and the probability is 1 p that it will be in class A c . Moreover, the outcomes of successive trials will be statistically independent. Therefore, if a particular sequence has x elements in A in n x elements A c , then, as a statistical outcome, its probability will be p x (1 p ) n x . 1

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D.S.G. POLLOCK: ECONOMETRICS There are altogether nC x = n ! / { ( n x )! x ! } such sequences, with x elements in A in n x elements in A c . These sequences represent a set of mutually exclusive ways in which the event in question can occur; and their probabilities
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Distribs - CHAPTER 10 Statistical Distributions In this...

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