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MIT OpenCourseWare http://ocw.mit.edu 14.30 Introduction to Statistical Methods in Economics Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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1 14.30 Introduction to Statistical Methods in Economics Lecture Notes 5 Konrad Menzel February 19, 2009 We distinguish 2 different types of random variables: discrete and continuous. Discrete Random Variables Definition 1 A random variable X has a discrete distribution if X can take on only a finite (or countably infinite) number of values ( x 1 ,x 2 ,... ) . Definition 2 If random variable X has a discrete distribution, the probability density function (p.d.f.) of X is defined as the function f X ( x ) = P ( X = x ) If { x 1 ,x 2 ,... } is the set of all possible values of X , then for any x / ∈ { x 1 ,x 2 ,... } , f X ( x ) = 0. Also f X ( x i ) = 1 i =1 The probability that X A for A R is P ( X A ) = f X ( x i ) x i A Example 1 If X is the number we rolled with a die, all integers 1 , 2 ,..., 6 are equally likely. More generally, we can define the discrete uniform distribution over the numbers x 1 ,x 2 ,...,x k by its p.d.f. 1 f X ( x ) = k if x ∈ { x 1 ,x 2 ,...,x k } 0 otherwise This corresponds to the simple probabilities for an experiment with sample space S = { x 1 ,x 2 ,...,x k } . Example 2 Suppose we toss 5 fair coins independently from another and define a random variable X which is equal to the observed number of heads. Then by our counting rules, n ( S ) = 2 5 = 32 , and 5 n (”k heads”) = , using the rule on combinations. Therefore k 5 1 1 5 1 5 P ( X = 0) = = , P ( X = 1) = = 0 32 32 1 32 32 , 5 1 10 5 1 10 P ( X = 2) = = , P ( X = 3) = = 2 32 32 3 32 32 , 5 1 5 5 1 1 P ( X = 4) = = , and P ( X = 5) = = . 4 32 32 5 32 32 1
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1 K ρ (X = X) 2 3 --- K-1 K X 1 Note that these probabilities sum to 1. P(X = X) 10 32 5 32 1 32 0 1 2 3 4 S X Note that in the die roll example, every single outcome corresponded to exactly one value of the random variable. In contrast for the five coin tosses there was a big difference in the number of outcomes corresponding to X = 2 compared to X = 0, say. So mapping outcomes into realizations of a random variable may lead to highly skewed distributions even though the underlying outcomes of the random experiment may all be equally likely. 1.1 The Binomial Distribution To generalize the preceding example, suppose we look at a sequence of n independent and identical trials, each of which can result in either a ”success” or a ”failure” (not necessarily with equal probability), and we are interested in the total number X of successes. Example 3 Suppose we sample 100 pieces from a batch of car parts at the producing plant for quality control. A piece passing the quality checks would constitute a ”success”, a piece falling short of one or more of the criteria would be a ”failure”. We are interested in the distribution of failures as a function of the total share of defective parts in the batch in order to infer from the sample whether we have good reason to believe that no more than, say, 1%
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