$X\sim U(0,1)$, $Y\sim U(0,2)$, 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?
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Is this under the assumption that $X$ and $Y$ are independent random variables? – user60887 Dec 10 '14 at 05:56
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If you know the marginal distributions and you know that they're independent, then you know the joint distribution. – Michael Hardy Dec 10 '14 at 05:58
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No, I don't know if they are independent or not, that's the problem – user133140 Dec 10 '14 at 05:59
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Then there is no way to determine the distribution of $T$. Intuitively this should be clear because if one doesn't know how $X$ and $Y$ vary with respect to each other then how can the probability that $T$ takes on a certain value be computed? – Sudarsan Dec 10 '14 at 06:01
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Notice the edit that changed $X$~$U(0,1)$ to $X\sim U(0,1)$. That is standard usage. ${}\qquad{}$ – Michael Hardy Dec 10 '14 at 06:11
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Ok, I got it, thanks again – user133140 Dec 10 '14 at 06:14
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$X\sim U(0,1)$, $Y\sim U(0,2)$, 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.
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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.
