Probability Theory Axioms and Distributions Cheat Sheet

Key Facts

• The probability axioms are $P(A) \ge 0$, $P(S) = 1$, and if $A_i$ are disjoint, then $P\left(\bigcup_i A_i\right) = \sum_i P(A_i)$.
• The complement rule is $P(A^c) = 1 - P(A)$.
• The addition rule for two events is $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
• Conditional probability is defined by $P(A \mid B) = \frac{P(A \cap B)}{P(B)}$ when $P(B) > 0$.
• Events $A$ and $B$ are independent if $P(A \cap B) = P(A)P(B)$, equivalently $P(A \mid B) = P(A)$ when $P(B) > 0$.
• Expected value and variance are $E[X] = \sum_x x p(x)$ for a discrete random variable and $\operatorname{Var}(X) = E[X^2] - (E[X])^2$.
• For a Binomial random variable $X \sim \operatorname{Bin}(n,p)$, $P(X=k) = \binom{n}{k}p^k(1-p)^{n-k}$, $E[X] = np$, and $\operatorname{Var}(X) = np(1-p)$.
• For a Normal random variable $X \sim N(\mu,\sigma^2)$, the standardized variable is $Z = \frac{X - \mu}{\sigma}$.

Vocabulary

Sample space

The sample space $S$ is the set of all possible outcomes of a random experiment.

Event

An event $A$ is a subset of the sample space whose probability $P(A)$ can be measured.

Conditional probability

Conditional probability $P(A \mid B)$ is the probability that event $A$ occurs given that event $B$ has occurred.

Independence

Two events are independent when knowing one occurred does not change the probability of the other.

Expected value

The expected value $E[X]$ is the long-run average value of a random variable.

Probability distribution

A probability distribution assigns probabilities to possible values of a random variable while totaling $1$.