# Question: Let W1 W2 be independent each with a Cauchy distribution

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 = W 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 = W is given by the convolution formula,

(d) Show that

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

(a) Show that the pdf of X1 = (1/2)W1 is

(b) Let Y1 = X1 + X2 = W 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 = W is given by the convolution formula,

(d) Show that

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

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