## Announce

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## \sigma-algebra

$$Pr(X|Y)=\frac{Pr(X \cap Y)}{P(Y)}$$ $$Pr(X \cap Y) = Pr(X|Y) P(Y)$$

## Basic

Conditional Expectation is a random variable (function of X), not yet value $$E[N|X] = E_N[N|X] = f(X) \text{ where } f(x) = E[N|X=x]$$

$$E[E[N|X]] = E_X[E_N[N|X]] = E[N]$$ Think of $E[E[N|X]] = \sum_x p(x) E[N|X=x]$ and $\sum_x p(x) p(n|x) = \sum_x P(n,x) = p(n)$

$$\text{Var}(X) = E[(X-EX)^2] = EX^2 - (EX)^2$$ $$\text{Var}(aX+bY) = a^2 \text{Var}(X) + 2ab \text{Cov}(X, Y) + b^2 \text{Var}(Y)$$

With $P(X \ge 0) = 1$ $$E[X] = \int_0^{\infty} P(X > \gamma) d\gamma$$