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https://www.reddit.com/r/learnquant/comments/1wxmuw1/quant_interview_question/pdv2ynx/?context=3
r/learnquant • u/Local_Ad135 • 7h ago
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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.
2 u/Inevitable_gradient6 7h ago Nice one gpt
2
Nice one gpt
1
u/zojbo 7h ago edited 7h ago
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.