Expected value for $a^x$

Question

Assume a random variable $X$. Assume, if this simplifies the problem, that $X$ can take only non negative integer values. Does there exist any relation between $E(X)$ and $E(a^X)$, where $a$ is an arbitrary value in the interval $[0,1]$ and by $a^X$ I mean the random variable that takes value $a^x$ with the same probability $p$ such that $X$ takes probability $x$?

Thanks!

We cannot find $E(a^X)$ just knowing $a$ and $E(X)$. Inequalities might be possible. — André Nicolas, Aug 20 '13 at 14:30
Indeed, this is just the moment generating function $M_X(t)$ where $t=\log a$. — Thomas Andrews, Aug 20 '13 at 14:33

score 2 · Answer 1 · answered Aug 20 '13 at 15:01

2

$$E(X) = \lim_{a \to 1} \dfrac{d}{da} E(a^X)$$

answered Aug 20 '13 at 15:01

Robert Israel

448,999

score 0 · Answer 2 · answered Aug 20 '13 at 14:43

There is a simple relation between $E[f(X)]$ and $E[X]$ for any convex function $f$, namely $f(E[X]) \leq E[f(X)]$. This holds for any random variable X and holds with strict inequality if X is not degenerate. Hence, since $a^x$ is convex, we have that $a^{E[X]} \leq E[a^X]$ or $E[X] \leq log_a(E[a^X])$.

Expected value for $a^x$

2 Answers2