Probability

Sample Space: entire set of possibilities for an experiment.
Event: subset of the sample space.
Probability Distribution: describes how probabilities are assigned to events.

\[ Var(X) = \mathbb{E}[(X - \mathbb{E}[X])^2] \\ Var(X) = \mathbb{E}[X^2] - (\mathbb{E}[X])^2 \]

If X & Y are independent random variables, then:

\[ Var(X + Y) = Var(X) + Var(Y) \\ Var(XY) = Var(X)Var(Y) + E[X]^2Var(Y) + E[Y]^2Var(X) \]

Discrete distributions

Continuous distributions

Central Limit Theorem

Given a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution, provided the population has a finite mean and variance.