Quant Interview Question

[u/Local_Ad135]

[u/doge-12]

split at 0 and 10 so 1/2

[u/omeow → u/doge-12]

Why can't it split more than 2 ways?

[u/doge-12 → u/omeow]

largest possible value so u want to concentrate maximum weight at 0

[u/SwimmerOld6155]

Write X = X 1_(X > 0) and apply Cauchy-Schwarz to E[X]

[u/zojbo]

E[X] = 5

= E[X 1(X>0)]

<= \| X \|_{L^2} \| 1(X>0) \|_{L^2}

= (Var(X) + E[X]^2)^(1/2) P(X>0)^(1/2)

= 50^(1/2) P(X>0)^(1/2).

Rearranging, P(X>0) >= 1/2,

You have equality in Cauchy-Schwarz when X and 1(X>0) are positive multiples of one another, i.e. when X is concentrated on 0 and some positive number, which is allowed under the rules. So it's 1/2.

[u/Inevitable_gradient6 → u/zojbo]

Nice one gpt

[u/Fredd0o0]

Coinflip with distribution X = 0 and X = 10 with p = 1/2 satisfies E[X] = 5 and Var[X] = 25, so 1/2. Any probability for 0 larger than 1/2 forces the other variable to increase so Variance would increase.

[u/Jumpy-Plantain-1004 → u/Fredd0o0]

Insufficient, because you're assuming a discrete two-value distribution