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So $X_1$ and $X_2$ are two independent uniform random variables and I am looking for the PDF of $Y,$ where $Y = |X_1 − X_2|.$ I have to find the CDF of $Z = X_1 - X_2$ first, then find the CDF of $Y = |Z|,$ and then finally take the derivative to find the PDF of $Y.$. I know this:

$$F_Y(y) = P(Y ≤ y) = P(X_1 − X_2 ≤ y).$$

I found the answer to this first part here: http://www.math.wm.edu/~leemis/chart/UDR/PDFs/StandarduniformStandardtriangular.pdf but I don't understand their bounds of integration.

I know this is a long question but any help would be appreciated. I've been staring at this problem for hours. A very detailed explanation would really help me. Apologies for not understanding how to type with MathJax.

BruceET
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Yijing Wang
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  • There is a portion of wikipedia on this https://en.wikipedia.org/wiki/Triangular_distribution – Isaac Browne Apr 05 '17 at 02:24
  • I edited your Question. Please take a look (a) to catch errors, (b) maybe to learn a bit of MathJax. Also, if you mean $\mathsf{Unif}(0,1)$ as in the link, then you should say 'standard' uniform in the question. – BruceET Apr 05 '17 at 02:31

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