Question: 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)
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
-1.png)
(b) Let Y1 = X1 + X2 = W and Y2 = X1, where X2 = (1/2)W2. Show that the joint pdf of Y1 and Y2 is
-2.png)
(c) Show that the pdf of Y1 = W is given by the convolution formula,
-3.png){kind=small}
(d) Show that
-4.png){kind=small}
That is, the pdf of W is the same as that of an individual W.
f(x) = (1+4f) g(yi.yz) = f(yi-y2)ryz).-oo
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a w 1 2x 1 and dw 1 dx 1 2 Thus b For x 2 y 1 y 2 x 1 y 2 J 1 Thus g y 1 y 2 f y 2 fy ... View full answer
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