Useful Theorems

Theorem 1: Given a probability space $S$ and a collection of $n$ events $A_1, A_2, \ldots, A_n$, the expected number of these events to occur is $\sum_{i=1}^n P(A_i)$.

Theorem 2: $P(T \geq 1) \leq \E(T)$.

Theorem 3 (Murphy’s Law): Given mutually independent events events $A_1, A_2, \ldots, A_n$, the $P(T=0) \leq e^{-\E(T)}$.

Theorem 4 (Product Rule for Expectations): For any independent random variables $R_1, R_2$ then $\E(R_1 R_2)=\E(R_1) \E(R_2)$.

Variance

Definition. The variance of a random variable $R$ is $$\text{Var}(R) = \E\left(\left(R-\E(R)\right)^2\right) = \E(R^2) - \E(R)^2$$

Definition. For a random variable $R$, the standard deviation of $R$ is $\sigma(R)=\sqrt{\text{Var}(R)}$

Theorem. If $R$ and $S$ are independent random variable, then $$\text{Var}(R+S) = \text{Var}(R)+\text{Var}(S)$$

Next Class

1. Exam!