# MATH230, Probability II

Undergraduate course, *Lancaster University, Department of Mathematics & Statistics*, 2017

My role: Graduate Teaching Assistant

Main topics:

Review of basic probability.

Random variables and their probability distribution functions. Probability mass functions and probability density function. Quantiles.

Expectation and variance of random variables; linearity of expectation. Higher-order moments.

Binomial, Poisson, uniform, geometric exponential. Review from first year and then further properties of the Poisson and exponential.

The gamma, normal, beta and chi-squared, and their inter-relationships and justification as probability models. The Cauchy and Student-$t$.

Joint distribution of vector random variables; that is, systems of two or more random variables, marginal and conditional distributions. Expectations and variances of vector variables. Moment generating functions.

Properties of linear combinations of random variables.

Transformations of random variables: motivation, univariate and bivariate methods.

Limit theory: convergence of variables, laws of large numbers, Central Limit Theorem.

Multivariate normal distribution.