Let X be a positive random variable; i.e. P (X ≤ 0) = 0.
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Let X be a positive random variable; i.e. P (X ≤ 0) = 0.

[From: ] [author: ] [Date: 13-10-23] [Hit: ]
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Determine the range of values of k ∈ R for which the following inequality holds:
E[X^k] ≥ [E(X)]^k.

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I think you need Jensen's inequality. If phi(x) is convex then E[ phi(X) ] ≥ phi( E[x] )

SO the question boils down to when is phi(x)=x^k for x>=0 convex.

Check the second derivative, phi ''(x)=k(k-1)x^k for x>=0 this would need to be nonnegative
(convex is the same as what a calc 1 book calls concave up)

So we need k(k-1)>=0. So, either k<0 or k>1
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