Let $X$ be a cotinuous random variable uniformly distributed over $[-10,10]$. Let $Y$ be a random variable with pdf $f_Y(y) = \frac{1}{40}\ln \frac{20}{|y|}, -20 \leq y \leq 20$.

$X$ and $Y$ ARE NOT independent. How do find the pdf of the sum $X+Y$?

I can't use convolution because of their non-independence.

How do I begin?

  • $\begingroup$ How do you know that X and Y are not independent ? What is the link between X and Y ? $\endgroup$ – Dark Jan 27 '15 at 19:15
  • $\begingroup$ before getting the pdf of $Y$, $Y$ is actually a function of $X$. Is there a way to check in/dependence? $\endgroup$ – cgo Jan 27 '15 at 19:18
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    $\begingroup$ What function then? Give all information you have. $\endgroup$ – drhab Jan 27 '15 at 19:22
  • $\begingroup$ So we want to find the sum $X+Y$, where $Y=2Z(c-X)$. $2Z \sim U(0,2)$ and $X\sim U(-10,10)$ as mentioned above. $\endgroup$ – cgo Jan 27 '15 at 19:27
  • $\begingroup$ You need to know what the dependence is exactly. That is, you need to know the joint distribution $\endgroup$ – Luis Mendo Jan 28 '15 at 16:35

Where you can start :

You have that : $Y = f(X)$ and you want the pdf of $X+Y$.

Let $\phi : \mathbb{R}^2 \to \mathbb{R}$ be a smooth bounded function.

Compute $\mathbb{E}[\phi(X+Y,X)]$. That should give you the joint pdf of $(X+Y,X)$. Then you should know how to get the pdf of $X+Y$.

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