Expected value of a transformation

Question

If $X$ is a continuous random variable with $EX = \mu < \infty$ and $Y = \exp(a|X|)$ for some $a > 0$ Is $EY < \infty$. How might one go about confirming this? Is knowing the distribution of $X$ necessary?

Answer 1

It is possible for all $a>0$ $$E(Y)=\infty$$

let $X\sim Symmetrized-log-normal(0,1)$

$$ f_X(x) = \frac{1}{2\sqrt{2\pi}|x|} e^{-\frac{1}{2} (\log |x|)^2} , \hspace{.5cm} x\in (-\infty , +\infty) $$ so $E(X)=0$ and the distribution of $|X|$ is log-normal. Log-normal_distribution in-hence $E(e^{a|X|})$ equals to Moments generation function of log-normal that is infinite for all $a>0$ existence-of-the-moment-generating-function

so for all $a>0$

$$E(e^{a|X|}) =\infty$$

Answer 2

It is possible for $ E[Y] \rightarrow \infty$

Let $ X= \pm 1.5 $ with probability $ \frac{1}{4} $ (each).
Let $ X= \pm 1.5^2 $ with probability $ \frac{1}{8} $ (each).
Let $ X= \pm 1.5^3 $ with probability $ \frac{1}{16} $ (each).
$\vdots$

Show that $ E[X] = 0 $. (Note, you have to show that the sum converges absolutely.)

Show that $ E[Y] \rightarrow \infty$.