Mathematical (Actuarial) Statistics:(Exam P/CT3/CS1)

Explore how probability underpins statistics by modeling data with random variables, understanding distributions and probability distribution functions, and applying these concepts to fields like finance, manufacturing, and science.

Explore the fundamentals of set theory, including elements, sets, subsets, null sets, complements, and the basics of intersection and union, with examples and notation for probability.

Explore how Venn diagrams visualize probability and set operations, using a dice sample space to illustrate subset, complement, intersection, and union, and solve the math and science exam problem.

Explore the fundamentals of probability, define the axioms, and derive the addition rule for events, including non mutually exclusive cases with a sample space, subsets, and a Venn diagram.

Explore conditional probability, the multiplication rule, and independence, and visualize probabilities with tree diagrams. Build toward Bayes' theorem and the law of total probability with worked examples.

Apply probability with Venn diagrams to find the probability of both lab work and referral from a primary care visit, using the complement to compute 0.05.

Use a two-circle Venn diagram to solve an actuarial probability question: compute liability (0.04), property (0.10), and liability but not property (0.01), then find the not-claim area as 0.89.

Uses a probability table to determine the share of patients with regular heartbeat and low blood pressure from given totals for high, low, and normal pressures and regular or irregular heartbeat.

Explore a difficult exam probability problem on airline punctuality, using conditional probability and the law of total probability to compute arrivals and departures on time, early, and late.

Explore random variables as the core of statistics, model discrete and continuous distributions, and apply calculus to derive probability density and cumulative distribution functions.

Explain random variables, especially discrete ones, with dice and coin examples, and connect the probability distribution function and cumulative distribution function, highlighting finite and countable infinite sets and their relationships.

Study continuous random variables, their probability density function, and integrals yielding P(a < x < b) with total area 1. Relate the CDF to pdf and note non-negativity and monotonicity.

Explore how to extract the mean, variance, and moments (including skewness) from probability distribution functions, and how linear transformations affect expectations, illustrated with a dice example.

Explore functions of a random variable, showing how to derive the distribution of y = g(x) from x's CDF and PDF, including monotonic or invertible cases, to compute moments.

Explore probability distributions, focusing on parameter and distribution, and distinguish discrete versus continuous types using their relationships, such as chi-squared with normal and geometric with negative binomial.

Explore discrete probability distributions, their probability distribution functions, means, and variances. Examine uniform, Bernoulli, binomial, hypergeometric, negative binomial, geometric, and Poisson distributions and their interconnections.

This lecture introduces continuous probability distributions, highlighting the beta and gamma families, with focus on the uniform, exponential, chi-squared, and normal approximations, and connections to Poisson processes and generating functions.

Explore generating functions as a powerful, easier alternative to calculus for manipulating distributions, deriving parameters like skewness, and extending to multidimensional expectations in statistics.

Unpack the probability, moment, and cumulant generating functions, and see how derivatives reveal moments and cumulants, for both discrete and continuous variables.

Use the moment generating function to find the expected value of 100 times 0.5^X, where X has an MGF. Compute with exponentials and logarithms to get about 41.9.

Explore joint distributions by combining multiple random variables to form data, derive marginal and conditional distributions, and examine correlation without assuming causation.

Explore joint probability functions by extending single-variable distributions to two dimensions, examining marginal and conditional distributions, convolutions, and covariance and correlation between x and y.

Explore a sample exam question on a joint exponential distribution for waiting times x and y, computing the probability that total time is under 60 minutes.

Explore conditional expectations by linking conditional distributions to mean and variance, and apply generating functions and moment generating functions to compound distributions in insurance.

Explore conditional expectations by deriving the mean of y given x (the regression of y on x) and proving that E[E[Y|X]] = E[Y].

Explore the variance decomposition in conditional expectations, proving that Var(Y) = E[Var(Y|X)] + Var(E[Y|X]); the lecture guides step-by-step through the proof and exam relevance.

Compute the mean and variance of total claims S in compound distributions by conditioning on the number of claims X and applying E[S|X] and Var(S|X) formulas, i.e., E[S]=E[E[S|X]] and Var(S)=E[Var(S|X)]+Var(E[S|X]).

Use moment generating functions to calculate the mean and variance of compound distributions. Recap cumulative generating function and derivatives around zero and around the mean for quick moment calculations.

Derive the moment generating function for the total claims S in a Poisson‑compound model and outline the mean and variance via differentiation of the mgf.

This lecture analyzes exam questions on conditional distributions, modeling claims with an exponential distribution and deductible to show how a 10% mean reduction affects the variance via the memoryless property.

Explore the central limit theorem and how the sample mean becomes a random variable with a normal distribution. Learn how this tool helps infer parameters and optimize processes using data.

Explore the central limit theorem's power and history from 1733 to Mandelbrot, noting its use in inference and the cautions needed for finance and risk management.

Apply the central limit theorem to sums of coin tosses and the normal approximation to binomial data, using standardization and tables to estimate probabilities.

Apply the continuity correction to the normal approximation of discrete distributions via the central limit theorem, using 0.5 half-unit bounds to include exact discrete values like x = 3.

Apply the central limit theorem to approximate the sample mean distribution for a discrete X with values 1, 2, 3 (probabilities 0.6, 0.3, 0.1), first for n=2, then n=50.

Explore sampling and inference by defining the statistic as a function on a random sample. Relate population parameters and distributions to an estimate, point estimation, confidence intervals, and hypothesis testing.

Turn data into information by linking population, sample, random variable, and observation to the statistic X bar as a mean estimator.

Explore the sample mean as a statistic and random variable; its expected value equals the population mean, its variance is sigma^2/n, the standard error is sigma/sqrt(n).

Learn how the expected value of the sample variance equals the population variance by using n-1, and why this bias correction was proposed by Bazil.

Explains why the sample variance uses 1 over n minus one, derives its variance via the chi-squared distribution, and notes independence of the sample mean and variance for normal population.

examine the t result as an adaptation of the central limit theorem, where a normal variable divided by a chi-squared term with df yields the t distribution.

Examine the f result and the f distribution, using the ratio of two chi-square variances from independent samples to set up the f test for comparing population variances.

Explore sampling distribution of the sample variance from the normal distribution. Link chi-square with n-1 degrees of freedom to the shape changes for small and large samples.

Explore point estimation in actuarial statistics, comparing the method of moments and the method of maximum likelihood, and treat estimators as random variables with their own distributions.

This lecture explains the point of point estimation: knowing the distribution and its parameters lets us answer probability questions, using moments and likelihood to estimate parameters ahead of goodness-of-fit checks.

the method of moments equates sample moments with population moments to estimate parameters, illustrated by using the Poisson sample mean for lambda and solving np and np(1-p) for binomial cases.

Explore the method of maximum likelihood, contrast it with the method of moments, and learn the four steps—likelihood, log-likelihood, derivative to zero, and verify maximum with the second derivative.

Explore the properties of estimators—bias, mean squared error, and consistency—by comparing method of moments and maximum likelihood, and understanding estimators as random variables with distributions.

Explain the Cramér–Rao lower bound for estimator variance, showing unbiased estimators cannot beat it, and connect to the asymptotic normal distribution of the maximum likelihood estimator via log-likelihood's second derivative.