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## Homework Statement

Prove that if X is a positive-valued RV, then E(X^k) ≥ E(X)^k for all k≥1

## The Attempt at a Solution

Why do I feel like this is a counter-example:

X = {1,2,4,8,16,...} (A positive-valued RV)

m(X) = {1/2,1/4,1/16,1/32,...} (A distribution function that sums to one)

Yet clearly,

[tex]E[X] = \sum _{k=1} ^{∞} \frac{1}{2^k} {2}^{k-1} = \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + \frac{1}{2} + ... = ∞[/tex]

So the expected value diverges (ie. doesn't exist). So I can't do the proof because for this RV, the expectation DNE.

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