Say, I have a function containing a random variable such as $ f(X)$, where $X $ is the random variable that comes from a family of random variables that differ only in the first and second moments (e.g. family of normal curves). Is it meaningful to understand how the moments of $ f(X)$ change when the a moment of $X $ changes, holding other moments of $X $ constant?
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$\begingroup$ Welcome to CV. It's hard to make sense of this, because $X$ is just "a random variable." What could it mean to change its variance? Evidently that would change $X$ into another random variable $Y$, but the concept of "derivative" requires that you be able to induce arbitrarily small changes in some well-defined sense. We therefore need more information just to get started understanding what you are trying to ask. $\endgroup$– whuber ♦Commented Mar 19, 2023 at 14:09
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$\begingroup$ OP: Does the distribution for $X$ have a scale parameter? If so, you could consider whether the partial derivative of the expectation of $f(X)$ with respect to that scale parameter is well-defined. This can sometimes be made easier by using the law of the unconscious statistician. $\endgroup$– GalenCommented Apr 26, 2023 at 20:17
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