qm: probability trees and conditional expectations

$$E(X)$$

$$E(X)=\sum_{i=1}^{n}P\left(X_{i}\right)X_{i}$$

$$\sigma^2\left(X\right)$$

$$\sigma^2\left(X\right)=E\left\lbrack\left(X-E\left(X\right)\right)^2\right\rbrack$$

$$==\sum_{i=1}^{n}P\left(X_{i}\right)\left(X_{i}-E\left(X\right)\right)^2$$

conditional expected values

$$E(X\vert S)=\sum_{i=1}^{n}P\left(X_{i}\vert S\right)X_{i}$$

total probability rule for the expected value

$$E(X)=E\left(X\vert S)P\left(S)+E(X\vert S^{c}\right)P\left(S^{c}\right)\right.$$

total probability rule

$P\left(A\right)=\sum_{n}^{}P\left(A\cap B_{n}\right)$

bayes formula

$$P\left(event\vert info\right)=\frac{P\left(info\vert event\right)}{P\left(info\right)}P\left(event\right)$$