+ ' 1. (24 pts) Let WW2 be independent, each with a Cauchy distribution. That is, their pdf's are given by 1 h(w) *(1 +

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1 24 Pts Let Ww2 Be Independent Each With A Cauchy Distribution That Is Their Pdf S Are Given By 1 H W 1 1 (30.14 KiB) Viewed 41 times

+ ' 1. (24 pts) Let WW2 be independent, each with a Cauchy distribution. That is, their pdf's are given by 1 h(w) *(1 + w) --00 < <00, i = 1,2 We would like to find the pdf of the sample mean, W = (Wi+W2)/2. a) Show that the pdf of X, = 0.5W, is 2 f(0) = -20 < x < 00 7(1 +42;) b) Let Yi = X1 + X2 = W and Y2 = X1, where X2=0.5W. Show that the joint pdf of Yi and Y. is 2 (1.72) = 10 - 1)/(wa) 7[1 +401 - ) (1 +423) -00 <<00,-00 < y2 <0. c) Show that the pdf of Y; = W is given by the convolution formula, + 2 o(vn) = [sin - )s(wady? d) Show that 91 (9) = *(1 +33) -00<41 <00. That is, the pdf of W is the same as that of an individual W.