- #1

joshmccraney

Gold Member

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- Homework Statement:
- Probability that ##Y>3X## given ##Y>0## where ##X,Y## are ##N(0,1)## random variables

- Relevant Equations:
- Nothing comes to mind

After plotting the above (not shown) I believe one way (the hard way) to solve this problem is to compute the following integral where ##f(x) = e^{-x^2/2}/\sqrt{2\pi}##: $$\frac{\int_0^\infty \int_{3X}^\infty f(X)f(Y)\, dydx + \int_{-\infty}^0 \int_0^\infty f(X)f(Y)\, dydx}{\int_{-\infty}^\infty \int_0^\infty f(X)f(Y)\, dydx}$$

But evidently there's a geometric approach that's much easier. Any help here?

To elaborate, I can see how quadrant II has angle ##\pi## and quadrant I has angle ##\arctan (1/3)## and the total region we consider is ##\pi## angle, so that the total angles considered divided by the angle allowed is the solution ##(\pi/2+\arctan(1/3))/\pi## but I don't understand how these angle additions relate to the normal distribution above.

But evidently there's a geometric approach that's much easier. Any help here?

To elaborate, I can see how quadrant II has angle ##\pi## and quadrant I has angle ##\arctan (1/3)## and the total region we consider is ##\pi## angle, so that the total angles considered divided by the angle allowed is the solution ##(\pi/2+\arctan(1/3))/\pi## but I don't understand how these angle additions relate to the normal distribution above.

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