Let W1, W2 be independent, each with a Cauchy distribution. In this exercise we find the
Chapter 5, Problem 5.2-12(choose chapter or problem)
Let W1, W2 be independent, each with a Cauchy distribution. In this exercise we find the pdf of the sample mean, (W1 + W2)/2. (a) Show that the pdf of X1 = (1/2)W1 is f(x1) = 2 1 + 4x2 1 , < x1 < . (b) Let Y1 = X1 + X2 = W and Y2 = X1, where X2 = (1/2)W2. Show that the joint pdf of Y1 and Y2 is g(y1, y2) = f(y1 y2)f(y2), < y1 < , < y2 < . (c) Show that the pdf of Y1 = W is given by the convolution formula, g1(y1) = f(y1 y2)f(y2) dy2. (d) Show that g1(y1) = 1 1 + y2 1 , < y1 < . That is, the pdf of W is the same as that of an individual W. 5.
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