We know that $X-2Y \sim N(0,5\sigma^2)$, but how to compute $P(X>|2Y|)$?
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2$\begingroup$ The distribution of X-2Y is not useful for answering the question. Look at the joint density of X and integrate over the region where X>2Y for Y positive and X>-2Y when Y is negative, Also add the self-study tag to your list of tags. $\endgroup$– Michael R. ChernickFeb 26, 2018 at 5:47
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$\begingroup$ @MichaelChernick: At first I got stuck at the integral part, and now I figured it out. Btw, I came across this question on the Internet and found it interesting. Does it belong to self-study question? $\endgroup$– NeedleFeb 27, 2018 at 20:19
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$\begingroup$ It still should be self study as it is a type of class exercise. $\endgroup$– Michael R. ChernickFeb 27, 2018 at 20:27
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2 Answers
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$P[X>|2Y|] = P[X>0]*P[X^2 > 4Y^2 | X > 0] = 0.5*P[\frac{Y^2}{X^2}<0.25]$
Now, note that $X^2$ and $Y^2$ are both squares of standard normal variates which means they follow $\chi^2_{(1)}$ distribution. Now you can use the result that if U and V are independent and follow chi squared distribution with degrees of freedom $a$ and $b$ respectively then the ratio $\frac{U}{V}$ follows F-Distribution with parameters $a$ and $b$ respectively.
Now to find $P[\frac{Y^2}{X^2}<0.25]$ use the tables or if you're not allowed to do that just integrate the pdf of F-Distribution over the specified range. The integration should not be much time consuming since the degrees of freedom are (1,1) hence the pdf will reduce to a relatively simple function.
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$\begingroup$ @Glen_b Yes I missed it, Can you give it a check now? $\endgroup$ Feb 26, 2018 at 9:14
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Here is a hint. Since $X$ and $Y$ are independent, their joint density is $f_{XY}(x,y)=f_X(x)f_Y(y)=\frac{1}{2\pi\sigma^2}\exp(-\frac{x^2+y^2}{2\sigma^2})$. Further, because of symmetry, $$ \text{Prob}[X>|2Y|] = 2\text{Prob}[X>2Y,Y>0] = 2 \int_0^{+\infty} \text{d}y \int_{2y}^{+\infty} \text{d}x f_{XY}(x,y)\,, $$ which you can easily solve if you transform to polar coordinates.
Another way of looking at it is to think of $f_{XY}(x,y)$ as a perfectly round cake out of which you take a wedge with angle $2\text{arctan}(\frac{1}{2})$. The probability mass of that wedge should therefore be $\text{arctan}(\frac{1}{2})/\pi \approx 0.4636/3.1415=0.1476$.
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$\begingroup$ I can't understand the second way. Could you give more explanation? $\endgroup$– NeedleFeb 27, 2018 at 4:56
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$\begingroup$ @Needle: Try to imagine what the area is in the $(x,y)$ plane where the event $X>|2Y|$ is true. It is a wedge with its tip at $(0,0)$ and an outward angle of $\alpha=2\text{arctan}(\frac{1}{2})$. The density $f_{XY}$ is rotation symmetric around $(0,0)$. Integrating $f_{XY}$ over the whole plane gives probability 1. Integrating over the wedge alone gives $\text{Prob}[X>|2Y|]$. $\endgroup$ Feb 27, 2018 at 8:53