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# Let W1,W2 be independent, each with a Cauchy distribution.

ISBN: 9780321923271 41

## Solution for problem 12E Chapter 5.2

Probability and Statistical Inference | 9th Edition

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Problem 12E

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

(b) Let Y1 = X1 + X2 =  and Y2 = X1, where X2 = (1/2)W2. Show that the joint pdf of Y1 and Y2 is

(c) Show that the pdf of Y1 =  is given by the convolution formula,

(d) Show that

That is, the pdf of  is the same as that of an individual W.

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Let W1,W2 be independent, each with a Cauchy distribution.

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