Let $X, Y$ be iid standard uniform variables, and let $T = X + Y$. The goal is to find the joint PDF of $X$ and $T$. The work I've done so far is to find the PDF of $T$ by evaluating the convolution, but I'm not even sure if that's useful because $X$ and $T$ aren't independent, so the joint PDF shouldn't be the product of the marginal PDFs. I thought about working from the CDF and writing it as $P(X \leq x, T \leq t) = P(X \leq x, X + Y \leq t)$. I didn't know where to go from there due to the lack of independence. Can anyone provide suggestions?
Joint PDF of 2 dependent variables w/ convolution
probabilityprobability distributionsrandom variablesuniform distribution
Best Answer
You are almost there. Note that $$ P(X\le x, T \le t) = P(X \le x, X+Y \le t) = P(X \le x, X \le t - Y), $$ It follows from $$ P(X \le x, X \le t - Y| Y = y) = \int_0^{\min\{x, t- y\}} du = \min\{x, t-y\} $$ that \begin{align*} &P(X \le x, X \le t - Y) \\ &= \int_0^1 P(X \le x, X \le t - Y| Y = y) dy \\ &= \int_0^1 \min\{x, t-y\} dy \\ &= \int_0^{\max\{t-x,0\}} x dy + \int_{\max\{t-x,0\}}^1 (t-y)dy \\ &= x\max\{t-x,0\} + (1-\max\{t-x,0\})t -\frac{1 - (\max\{t-x,0\})^2}{2}. \end{align*}