Probability

Introductory Probability as taught in Sheldon Ross' book

Expectation and Variance

Some definitions and properties.

Total Expection

Dr. Xu refers to this as the smooth property. \(\displaystyle E(X) = E(E(X|Y))\)

Proof

Total Variance

This one is not used as often on tests as total expectation \(\displaystyle Var(Y) = E(Var(Y|X)) + Var(E(Y | X)\)

Proof

Delta Method

See Wikipedia

Relationships between distributions

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

Poisson Distributions

Sum of poission is poisson sum of lambda.

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 \(\displaystyle X_2 + X_2 \sim \Gamma(k_1 + k_2, \theta)\). If \(\displaystyle X_1 \sim \Gamma(\alpha, \theta)\) and \(\displaystyle X_2 \sim \Gamma(\beta, \theta)\), then \(\displaystyle \frac{X_1}{X_1 + X_2} \sim B(\alpha, \beta)\)

T-distribution

Ratio of normal and squared-root of Chi-sq distribution yields T-distribution.

Chi-Sq Distribution

The ratio of two normalized Chi-sq is an F-distributions

F Distribution

Too many. See the Wikipedia Page. Most important are Chi-sq and T distribution