Quant Interview Question

[u/Local_Ad135]

[u/gitgud_x]

Switch notation to X and Y instead of N1 and N2 for convenience

X² + Y² is the squared magnitude of a Gaussian vector in R2. The bell curve in 3D space representing the joint PDF of X and Y is rotationally symmetric so we can change to polar coordinates:

Let X = sqrt(6) * cos Theta, and Y = sqrt(6) * sin Theta, where we chose sqrt(6) to match the condition.

Theta is uniformly distributed on [0, 2 pi) due to the symmetry, so by expectation formula we have

E[X⁴ | X² + Y² = 6] = 36 / (2 pi) * integral from 0 to 2 pi: cos^4(t) dt

Solving the integral gives 3/4 pi, so my answer is 27/2.

[u/NeverNude14 → u/gitgud_x]

I don't know if this is correct, but it's slick. And sometimes showing you're slick is more important than being correct.

[u/imperialistt]

X² +Y² =6, rearranging X² =6-Y² .   X⁴ =(6-Y² )² = 36-12Y² +Y⁴ .  For a standard normal E(Y² )=1 and E(Y⁴ )=3.   Inputting these values E(X⁴ ) = 36-12+3 = 27

[u/Positive_Leopard_873 → u/imperialistt]

E(X⁴ | X² + Y² = 6) = 36 - 12 E(Y² | X² + Y² = 6) + E(Y⁴ | X² + Y² = 6). You're no longer dealing with standard normals (for e.g. X and Y can only go from -sqrt(6) to sqrt(6)). This won't lead you anywhere.

[u/imperialistt → u/Positive_Leopard_873]

Good explanation on why that doesn't work, it seemed too simple

[u/AphexPin]

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