6 - MIT OpenCourseWare http/ocw.mit.edu 14.30 Introduction...

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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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Problem Set #6 - Solution 14.30 - Intro. to Statistical Methods in Economics Instructor: Konrad Menzel Due: Tuesday, April 7, 2009 Question One Let X be a random variable that is uniformly distributed on [O,1] (i.e. f (x) = 1 on that interval and zero elsewhere). In Problem Set #4, you use the "2-stepV/CDF technique and the transformation method to determine the PDF of each of the following transformations, Y = g(X). Now that you have the PDFs, compute (a) IE[g(X)], (b) g(E[X]),(c) Var(g(X)) and (d) g(Var(X)) for each of the following transformations: 1. Y = xi, fv(y) = 4y3 on [O, 11 and zero otherwise. Solution to 1: We compute the four components: 4 51-4-080 (a) E[g(X)] =Gy(4y3)dy= (5y)0- - . orwecancomputeit using X: 4 51-4 E[g(X)] = J,' xidx = (5~4)~ - 5. (b) g(IE[X]) = (Sixdx)a = = & = 0.84. (c) The variance uses the result in part (a) (d) We need to compute Var(X) first:
And then transform it: g(Var(X)) = (A)' = & = 0.537 2. Y = ePx, fY(y) = on [i, 11 and zero otherwise. a Solution to 2: We compute the four components: (a) E[g(X)] = J: ~(t)d~ = = 1 - = 0.632 or we can compute it using X: e E[g(X)] = J,' ePxdx = (-epX)h = -; + 1. (b) g(E[X]) = e-(lixdx) = e-(i) = 0.607. (c) The variance uses the result in part (a), ij -- E[Y] = 1 - a, (d) Using Var(X) from part (a), Var(X) = &, we apply g(-): g(Var(X)) = e-A = 0.920. X 1 3. Y = 1 - e , fy (9) = on [O,1- :] and zero otherwise. 1-Y

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(a) We need to do a little more algebra for this problem: 1 or we can compute it using X: E[g(X)] = & (1 - epx)dx = (x + e-")A = 1 l+--l=L. e e (b) g(IE[X]) = 1 - e-(Jhxdx) = 1 - e-(i) = 0.393. (c) The variance uses the result in part (a), y = IE[Y] = i, combined with one of the identities for the variance: 1-1 Y lpy 1 dy-- 1 = J ( y - 1-3 1-Y ) e2 (d) Using Var(X) from part (a), Var(X) = A, we apply g(.): g(Var(X)) = 1 - e-A = 0.080.
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6 - MIT OpenCourseWare http/ocw.mit.edu 14.30 Introduction...

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