X and Y are independent standard normals. What is P(X/Y > 2), and what goes wrong if you try to compute E[X/Y]?

X and Y are independent standard normals. What is P(X/Y > 2), and what goes wrong if you try to compute E[X/Y]?

Approach: Write the pair in polar coordinates, where rotational symmetry makes the angle uniform. That identifies the distribution of the ratio directly and gives its tail in closed form.

0.1476. In polar coordinates the joint density has rotational symmetry, so the angle is uniform on [0, 2 pi) and X/Y is the cotangent of a uniform angle. That makes the ratio a standard Cauchy distribution with density 1/(pi(1 + t^2)) and CDF 1/2 + arctan(t)/pi. Hence P(X/Y > 2) = 1/2 - arctan(2)/pi = 0.5 - 0.3524 = 0.1476. The mean does not exist: the divergent mean integral ∫ |t|/(pi(1 + t^2)) dt grows logarithmically. The practical consequence is that the sample mean of n Cauchy draws has exactly the same distribution as a single draw, so averaging ratios buys nothing.

Follow-up: What is the distribution of the ratio of two independent exponentials with the same rate?

Key concepts: cauchy distribution, polar coordinates, rotational symmetry, divergent mean integral.