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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 9 Konrad Menzel March 10, 2009 Functions of Random Variables In this part of the lecture we are going to look at functions of random variables, Y = u ( X ). Note that Y is again a random variable: since X is a mapping from the sample space S into the real numbers, X : S R → and u : R R , the composition of u and X is also a mapping from S into the real numbers: → Y = u X : S R ◦ → Example 1 If X is the life of the first spark plug in a lawnmower, and Y the life of the second, we may be interested in the sum of the two, Z = X + Y . Example 2 Before coming to MIT, I applied for several German fellowships, so I would receive a monthly stipend of X Euros, depending on which fellowship I was going to get, and the exchange rate in, say, September 2005 was going to be Y Dollars per Euro. Each quantity was uncertain at the time I was applying, but since I was going to spend the money in the US, the main quantity of interest was the dollar amount Z = X ∗ Y I was going to receive (at least in terms of the Dollar exchange rate, I could not complain). We now want to know how to obtain the density and c.d.f. for the transformed random variable u ( X ), so we can treat each problem involving a function of a random variable in the same way as a question involving only the random variable itself with a known p.d.f. We’ll consider three cases: 1. underlying variable X discrete 2. underlying variable X continuous 3. X continuous and u ( X ) strictly increasing The last case is of course a special case of the second, but we’ll see that it’s much easier to work with. 1 1.1 Discrete Case- ”2-Step” Method If X is a discrete random variable with p.d.f. f X ( x ), and Y = u ( X ), where u ( ) is a known deterministic · function. Then f Y ( y ) = P ( Y = y ) = P ( u ( X ) = y ) = f X ( x ) x : u ( x )= y Example 3 if 5 f X ( x ) = 1 x ∈ {− 2 , − 1 , , 1 , 2 } 0 otherwise Then if Y = g ( X ) = X , | | ⎧ ⎪ f X (0) = 1 if y = 0 ⎪ 5 f Y ( y ) = ⎨ f X ( − 1) + f X (1) = 5 2 if y = 1 ⎪...

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