Calculating $\mathbb{E}[Re^{iX}]$ where $R$ and $X$ are independent random variables.

Question

Let $R$ be an exponential random variable with parameter $\lambda>0$ and let X be independent from R and uniformly distributed on $(0,2\pi)$, find $\mathbb{E}[Re^{iX}]$. My intuition tells me that the expectation of the product is the product of the expectations and that that the final answer should be $-\mathbb{E}[R]$ since $\mathbb{E}[X]=\pi$. But using Euler's formula and integrating I get that $\mathbb{E}[e^{iX}]=0$ which doesn't make sense. Any tips on this would be appreciated.

Answer 1

Your wrong assumption seems to be that $\mathbb Ee^{iX}=e^{\mathbb EiX}=e^{i\pi}=-1$.

The first equality is not correct.

It is correct that $\mathbb Ee^{iX}=0$.

Observe that $e^{iX}$ and $-e^{iX}=e^{i(X+\pi)}$ will have equal distribution (both uniform on unit circle).

This tells us that $-\mathbb Ee^{iX}=\mathbb Ee^{iX}$ or equivalently $\mathbb Ee^{iX}=0$.