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So here's the following question and I'll explain my solution so far, just want to know if I'm on the right track.

Let X, Y : Ω → R be independent random variables on a probability space (Ω, F, P) such that X is distributed according to the beta distribution β2,1, and Y according to the exponential distribution E2. Determine the probability that X > 2Y .

So I know that P (X>2Y) is the same as saying P (X-2Y >0). So I figured I could just find the density of X-2Y and thus it's CDF. I used the fact that they are independent to show that their joint density is: $P (x,y) = 4xe^{-2y}$. (Sorry for no latex abilities).

I then integrated this over the intervals (0,$\infty$) dy and (0,(c+2y)) dx (where c is some constant such that x < c-2y) to find that the density is equal to 2(c^2 + c + 1).

Where exactly do I go from here? Can I simply integrate again wrt z where z=x-2y? Or have I just messed it up entirely?

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    $\begingroup$ I have edited in some LaTeX and with that as a guide you can probably do the rest. $\endgroup$ Commented Mar 21, 2018 at 13:09

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If you are already integrating over the joint density, bringing in an additional variable $c$ is just adding unnecessary complication to a simple problem. Given the joint density $f_{X,Y}$ for random variables $0 \leqslant X \leqslant 1$ and $Y \geqslant 0$ you obtain the probability of your event by integration over the joint density:

$$\mathbb{P}(X > 2Y) = \int \limits_0^1 \int \limits_0^{x/2} f_{X,Y}(x,y) dy dx.$$

Try using your joint density to evaluate this probability, and then compare your result to empirical tests using comparisons simulated values from the specified marginal distributions.

Also, spend an hour to learn some basic Latex!

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  • $\begingroup$ (For some very basic $\LaTeX$ 15 minutes is enough) $\endgroup$ Commented Jul 28, 2024 at 22:58

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