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Let $Y=1/4(X_1 + X_2 + X_3 + X_4)$, where $X_1$, $X_2$, $X_3$ and $X_4$ are i.i.d. r.v.s (independent and identically distributed random variables) with a Cauchy pdf $$f_X(x) = \frac{a}{\pi(x^2 + a^2)}$$ I need to solve the characteristic function of Y and pdf of Y. I cannot apply Cauchy pdf to this problem, can you please give me directions to solve this problem?

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  • $\begingroup$ I typeset your question with Latex. Please check that everything is as you intended. $\endgroup$ – rubik Jan 20 '15 at 9:21
  • $\begingroup$ I checked, thanks $\endgroup$ – mtarim Jan 20 '15 at 9:30
  • $\begingroup$ Welcome to our site! $\endgroup$ – kjetil b halvorsen Jan 20 '15 at 9:32
  • $\begingroup$ NB: @rubik, it's not LaTeX, it's MathJax :) $\endgroup$ – Shaun Jan 20 '15 at 10:11
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    $\begingroup$ @Shaun yes and what is MathJax? An implementation of TeX/LaTeX for the browser, in Javascript. $\endgroup$ – rubik Jan 20 '15 at 13:13
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Hint:

Find the characteristic function of $Y$ on base of: $$\phi_{Y}\left(t\right)=\mathbb{E}e^{itY}=\phi_{X}\left(\frac{t}{4}\right)^{4}$$ and have a good look at it. Most probably you will recognize this function and the PDF of $Y$ will be evident then.

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Since the characteristic function of $X_1$ is $e^{-a|t|}$, the characteristic function of $S=X_1+X_2+X_3+X_4$ is $e^{-4a|t|}$, so $S$ is still Cauchy distributed and the pdf of $S$ is given by: $$ f_S(x) = \frac{4a}{\pi(x^2+(4a)^2))}.$$ This gives that the arithmetic mean of $X_1,X_2,X_3,X_4$ has the same distribution of $X_1$.

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  • $\begingroup$ i'm not familiar with this formula neither, can you please iterate one step further? $\endgroup$ – mtarim Jan 20 '15 at 12:02

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