# MATH230, Probability II

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

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.