$X\sim U(0,1)$, $Y\sim U(0,2)$, how can I find CDF of $T=X+Y$ without knowing the joint PDF of $X$ and $Y$? $X\sim U(0,1)$, $Y\sim U(0,2)$, how can I find CDF of $T=X+Y$ without knowing the joint PDF of $X$ and $Y$?
Does anyone could help me with this?
 A: QUOTE
$X\sim U(0,1)$, $Y\sim U(0,2)$, how can I find CDF of $T=X+Y$ without knowing the joint PDF of $X$ and $Y$?
END OF QUOTE
If $X\sim U(0,1)$ and $Y=2X$ then $Y\sim U(0,2)$ and $X+Y\sim U(0,3)$, and this has a standard deviation $3$ times as big as that of $X$. But if $X\sim U(0,1)$ and $Y\sim U(0,2)$ and $X$ and $Y$ are independent, then $X+Y$ has a standard deviation only $\sqrt{5}\approx2.236$ times as big as that of $X$ (since $\sqrt{1^2+2^2}=\sqrt 5$).  So there is more than one thing that the distribution of $X+Y$ could be, consistenly with the given information about the marginal distributions of $X$ and $Y$.
In other words, you cannot know the distribution of $X+Y$ without further information beyond the two marginal distributions.
A: There's no real way of tackling this problem without the joint PDF being used indirectly or directly. Luckily, the joint PDF is easy to find because you're given independent random variables; just multiply the two PDFs for $X$ and $Y$ to get the joint PDF.
One you have that, you'll need to do some integration to find the CDF. Be sure to keep track of supports as you do the integration.
