How to calculate the expectation of a complex random variable the distribution of which is a function of another random variable?

Question

There is a complex random variable $ X \sim \cal CN(0,10^{-0.1 \rm Y})$, and also $Y \sim \cal N(0,\sigma^2)$ is a real random variable. How can I calculate the expectation $\mathbb E (|X|) $, where $| \cdot |$ means 'absolute value'. If $Y$ is a determined number ,then $|X|$ would follow the Rayleigh distribution, but I don't know what it would be in this case. Is there any textbook about how to solve such problem? Any comments would help! Thanks a lot!

Answer 1

Hint: $X \sim \cal CN(0,10^{-0.1 \rm Y})$ actually means you know the distribution of $X$ conditioned on $Y$. That is, you know $ X \mid Y \sim \cal CN(0,10^{-0.1 \rm Y})$

Then apply the tower property $E[Z] = E [ E [ Z | Y ]]$ for $Z=|X|$.