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tags: Probability
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# Further Topics on RVs: Conditional Expectation and Variance Revisited

by AWfulsome

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Two useful probability laws:
* Law of iterated expectations
$$
E[E[X|Y]] = E[X]
$$
* Law of total variance
$$
\mathrm{var}(X) = E[\mathrm{var}(X|Y)] + \mathrm{var}(E[X|Y])
$$

### More on Conditional Expectation

* $E[X|Y = y]$ in fact can be viewed as *a function of $Y$*, because its value depends on the value $y$ of $Y$.
* Is $E[X|Y]$ a random variable?
* What is the expected value of $E[X|Y]$?

:::spoiler Example
:::