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Unformatted text preview: 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 . 1 14.30 Introduction to Statistical Methods in Economics Lecture Notes 6 Konrad Menzel February 24, 2009 Examples Suppose that a random variable is such that on some interval [ a,b ] on the real axis, the probability of X belonging to some subinterval [ a ,b ] (where a ≤ a ≤ b ≤ b ) is proportional to the length of that subinterval. Definition 1 A random variable X is uniformly distributed on the interval [ a,b ] , a < b , if it has the probability density function 1 if a ≤ x ≤ b f X ( x ) = b − a 0 otherwise In symbols, we then write X ∼ U [ a,b ] F(x) x y a 1 b-a b Figure 1: p.d.f for a Uniform Random Variable, X ∼ [ a,b ] For example, if X ∼ U [0 , 10], then 4 4 1 1 P (3 < X < 4) = f ( t ) dt = dt = 10 10 3 3 What is P (3 ≤ X ≤ 4)? Since the probability that P ( X = 3) = 0 = P ( X = 4), this is the same as P (3 < X < 4). 1 Image by MIT OpenCourseWare. Example 1 Suppose X has p.d.f. ax 2 if 0 < x < 3 f X ( x ) = 0 otherwise What does a have to be? - since P ( X ∈ R ) = 1 , the density has to integrate to 1, so a must be such that ∞ 3 2 ax 3 3 27 1 = f X ( t ) dt = at dt = = a − 0 = 9 a 3 3 −∞ Therefore, a = 9 1 . What is P (1 < X < 2) ? - let’s calculate the integral 2 t 2 2 3 1 3 7 P (1 < X < 2) = dt = − = 9 9 · 3 9 · 3 27 1 What is P (1 < X ) ? ∞ 3 t 2 27 − 1 26 P (1 < X ) = f X ( t ) dt = dt = = 9 27 27 1 1 1.1 Mixed Random Variables/Distributions Many kinds of real-world data exhibit point masses at some values mainly for two different reasons: • some outcomes are restricted to certain values mechanically, so a lot of probability mass tends to cumulate right at the corners of the range of the random variable, e.g. daily rainfall can possibly take any positive real value, but there are many days at which rainfall is zero. • individuals taking economic decisions may respond to certain institutional rules by positioning themselves right at some kind of kinks or discontinuities, e.g. if we look at incomes reported to Social Security or the Internal Revenue Service, we observe ”bunching” of individuals at the top ends of the tax brackets (since for those individuals, a small increase in income would mean a discrete jump in the tax rate). The corresponding distributions are, strictly speaking, not continuous , because even though realizations can be any real-valued numbers, we can’t define a probability density function as we did in the previous section, but we’ll have to deal with the point masses separately. Some of this is going to come up in your econometrics classes, but we won’t spend time on this for now and only look at one example....
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six - MIT OpenCourseWare http://ocw.mit.edu 14.30...

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