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?

  • 2
    $\begingroup$ $\newcommand{\E}{\mathbb{E}}$Search up "law of total expectation". For example, if $X$ has density $f_X$, then $\E[Y]=\int \E[Y\mid X=x]f_X(x)\, dx$. $\endgroup$ – Minus One-Twelfth May 22 at 10:25

Use the fact that:

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

  • $\begingroup$ and $E(Y|X)=E(Y|X=\cdot)\circ X$. $\endgroup$ – Gabriel Romon May 22 at 10:50
  • 1
    $\begingroup$ I am not sure I understand the notation you have used. $\endgroup$ – Vizag May 22 at 11:00

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