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Suppose $X,Y$ are independent random variables and have density function $U(0,1)$, how can calculate probability density function of $W=|Y-X|\,, 0<w<1$.

A.D
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hadisanji
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1 Answers1

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For $0<w<1,$

The DF of $W$ is $$\begin{align}F_W(w) &= P[W \le w] \\&=P[|Y-X|\le w]\\&=P[-w \le Y-X \le w] \\&=P[(X,Y)\in A]\end{align}$$ where $$\begin{align}A &=\{ (x,y):-w \le y-x \le w\} \\&=\text{Area of A} \\ &=1-(1-w)^2\end{align}$$.

Hence $$F_W(w)= \begin{cases} 0, & w\le0 \\ 1-(1-w)^2,& 0<w<1 \\ 1, &w\ge1\end{cases}$$

Hence you can find the pdf.

A.D
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