I'm looking for an upper bound on $E(X^k)$ where $X$ is a random variable with $E(X)=1$. How can I go about doing this?
2025-01-12 23:55:08.1736726108
Upper bound on an expectation
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If $k >1$, we can consider $X = x^\frac{-1}{k}$ on $[0,1]$ with the Lebesgue measure.
Then we have $$\mathbb{E} X = \int_0^1 x^\frac{-1}{k} \textrm{d}x = \frac{k}{k-1} $$ So $X$ can be normalized, however $$\mathbb{E}X^k = \int_0^1 \frac{1}{x} \textrm{d}x = \infty$$