Introduction to Probability, Second Edition, 2nd Edition

9.7    Recap

To calculate an unconditional expectation, we can divide up the sample space and use the law of total expectation

$E\left(Y\right)=\sum _{i=1}^{n}E\left(Y|A_i\right)P\left(A_i\right),$

FIGURE 9.9PMF of $Y~\text{NBin}\left(r_0,b_0∕\left(b_0+t\right)\right)$ in gray and $Ỹ~\text{Pois}\left(r_{0}t∕b_0\right)$ in black, where $r_0=5$, $b_0=1$, $t=2$.

but we must be careful not to destroy information in subsequent steps (such as by forgetting in the midst of a long calculation to condition on something that needs to be conditioned on). In problems with a recursive structure, we can also use first-step analysis for expectations.

The conditional expectation $E\left(Y|X\right)$ and conditional variance $\text{Var}\left(Y|X\right)$ are random variables that are functions of X; they ...