Probability

Introductory Probability as taught in Sheldon Ross' book

Common Convolutions

This is important for tests.
See Relationships among probability distributions.

Normal Distributions

If \(\displaystyle X_1 \sim N(\mu_1, \sigma_1^2)\) and \(\displaystyle X_2 \sim N(\mu_2, \sigma_2^2)\) then \(\displaystyle \lambda_1 X_1 + \lambda_2 X_2 \sim N(\lambda_1 \mu_1 + \lambda_2 X_2, \lambda_1^2 \sigma_1^2 + \lambda_2^2 + \sigma_2^2)\) for any \(\displaystyle \lambda_1, \lambda_2 \in \mathbb{R}\)

Gamma Distributions

Note exponential distributions are also Gamma distrubitions If \(\displaystyle X \sim \Gamma(k, \theta)\) then \(\displaystyle \lambda X \sim \Gamma(k, c\theta)\).
If \(\displaystyle X_1 \sim \Gamma(k_1, \theta)\) and \(\displaystyle X_2 \sim \Gamma(k_2, \theta)\) then \(\displaystyle X_2 + X_2 \sim \Gamma(k_1 + k_2, \theta)\).

Gamma and Beta

If \(\displaystyle X_1 \sim \Gamma(\alpha, \theta)\) and \(\displaystyle X_2 \sim \Gamma(\beta, \theta)\), then \(\displaystyle \frac{X_1}{X_1 + X_2} \sim B(\alpha, \beta)\)