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

Chapter 5, Problem 12E

(choose chapter or problem)

Let \(W_{1}, W_{2}\) be independent, each with a Cauchy distribution. In this exercise we find the pdf of the sample mean, $\left(W_{1}+W_{2}\right) / 2\).

(a) Show that the pdf of \(X_{1}=(1 / 2) W_{1}\) is

\(f\left(x_{1}\right)=\frac{2}{\pi\left(1+4 x_{1}^{2}\right)}, \quad-\infty<x_{1}<\infty\)

(b) Let \(Y_{1}=X_{1}+X_{2}=\bar{W}\) and \(Y_{2}=X_{1}\), where \(X_{2}=(1 / 2) W_{2}\). Show that the joint pdf of \(Y_{1}\) and \(Y_{2}\) is \(g\left(y_{1}, y_{2}\right)=f\left(y_{1}-y_{2}\right) f\left(y_{2}\right), \quad-\infty<y_{1}<\infty\),

\(-\infty<y_{2}<\infty\)

(c) Show that the pdf of \(Y_{1}=\bar{W}\) is given by the convolution formula,

\(g_{1}\left(y_{1}\right)=\int_{-\infty}^{\infty} f\left(y_{1}-y_{2}\right) f\left(y_{2}\right) d y_{2}\).

(d) Show that

\(g_{1}\left(y_{1}\right)=\frac{1}{\pi\left(1+y_{1}^{2}\right)}, \quad-\infty<y_{1}<\infty\)

That is, the pdf of \(\bar{W}\) is the same as that of an individual \(W\)

Equation Transcription:

  

 

 

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Text Transcription:

W_1,W_2

(W_1+W_2)/2    

X_1=(1/2)W_1

f(x_1)=2/pi(1+4x_1^2), -infinity <x1< infinity

Y_1=X_1+X_2= bar W  

Y_2=X_1

X_2=(1/2)W_2.  

Y_1  

Y_2

g(y_1,y_2=f(y_1-y_2)f(y_2), - infinity <y_1<

-infinity<y2<infinity  

Y_1=bar W  

g_1(y_1)= int_-infinity^infinity  (fy_1-y_2)f(y_2)dy_2

g_1(y_1)=1/pi(1+y_1^2), -infinity< y_1<infinity

Bar W  

W