# Finding expected value when conditional distribution is known

If the distribution of $$Y$$ conditional on $$X=x$$ is known, and the distribution of $$X$$ is known, what would be the general process for finding the expected value $$\Bbb E[Y]$$? IS there a general process, or does one need to know the exact distributions?

• $\newcommand{\E}{\mathbb{E}}$Search up "law of total expectation". For example, if $X$ has density $f_X$, then $\E[Y]=\int \E[Y\mid X=x]f_X(x)\, dx$. – Minus One-Twelfth May 22 at 10:25

Use the fact that:

$$E[Y] = E[E[Y|X]]$$

• and $E(Y|X)=E(Y|X=\cdot)\circ X$. – Gabriel Romon May 22 at 10:50
• I am not sure I understand the notation you have used. – Vizag May 22 at 11:00