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Suppose we have a joint probability density function $f(x,y)$, how do we find the cumulative distribution function of $ T = X + Y $ if the support is $$ \begin{cases} 0 < X < 1 \\ 0 < Y < 1 \end{cases} $$ My thought is to discuss the cases when

$ 0 < t < 1 $ , I just integrate over a triangular region. My integral is: $ Pr(T < t) = \int_0^t \int_0^{t-y}f(x,y) ~dx ~dy $

$ 1 < t < 2 $ , I feel difficult about this one because when I draw the graph I found the region is a trapezoid.

So, am I on the right track? If so, did I do the 1st case right, and how do I integrate the trapezoid in the 2nd case?

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up vote 2 down vote accepted

Hint: Your procedure for $0\le t\le 1$ is correct. And you are very much on the right track, since you have drawn a picture!

For $1\lt t\le 2$, integrating over the trapezoid will work. To do the integration, you will either have to make a cange of variables, or (simpler) break up the region of integration into two parts.

But here is an easier way. In the picture for $t\gt 1$, notice the triangle at the upper right corner? Integrate over that. The limits of integration are then not hard to write down. But note that if you integrate first with respect to $y$, then $x$ travels from $t-1$ to $1$.

You will get the probability $p(t)$ that $T\ge t$. The probability that $T\le t$ is then $1-p(t)$.

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