Determine the range of values of k ∈ R for which the following inequality holds:
E[X^k] ≥ [E(X)]^k.
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
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