# Variance of a distribution's product with itself

I have a distribution $X$. By playing around with random samples from $X$, I've determined that $Var(X^i) > Var(X)$ where $i > 1$. However, I can't seem to find a formula for the expected variance of $X^i$, or why it should be greater than $Var(X)$.

Moving away from normal distributions, should it generalize that the scale parameter of any $X^i$ will be greater than $X$?

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$Var(X^i)$ = $\mathbb{E}[(X^i - \mathbb{E}[X^i])^2]$ = $\mathbb{E}[X^{2i}] - (\mathbb{E}[X^i])^2$ by definition. This expresses the variance of $X^i$ in terms of moments of $X$.

The generalization is false, because $\mathbb{E}[(\lambda X)^{2i}]$ = $|\lambda|^{2i}\mathbb{E}[X^{2i}]$ implies that the scale parameter of $(\lambda X)^i$ will be smaller than the scale parameter of $\lambda X$ when $\lambda$ is sufficiently close to zero and $i \gt 1$.

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Quick note, you may find useful discussion of why the formula for estimating the standard deviation for a sample uses $n-1$ as opposed to $n$.

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