Express $\mathbb{E}[f(X)]$ as a function of $\mathbb{E}[X]$?

Question

Let $X$ be a non negative random variable, $F$ its cdf and $\mu$ its expectation. Let $f:\mathbb{R}_+\to \mathbb{R}$. Is it possible to express $\mathbb{E}[f(X)]$ as a function of $\mu$ and not of $X$?

Answer 1

No, it's not possible to find $E[f(X)]$ given only $f$ and $E[X]$.

As an example, consider the two random variables $X_1$ and $X_2$, where $X_1$ is uniform on $[1,2]$ and $X_2$ is uniform on $[0,3]$. Then they have the same expectation of $1.5$. If we now take the function $f(X)=X^2$, we get that $$ E[X_1^2]=\frac{7}3\\ E[X_2^2]=\frac{31}{12} $$ Thus from $E[X]=1.5$ and $f(X)=X^2$, we have found two possible values for $E[f(X)]$, and thus we clearly need more information about $X$ before we can conclude. In particular, for this specific $f$, we have $$ E[X^2]=\operatorname{Var}(X)+(E[X])^2 $$ meaning we need to know the variance of $X$. For other $f$ we may need other information.