#### Description

Stat 400 is an introductory course to probability, the mathematical theory of randomness, and to statistics, the mathematical science of data analysis and analysis in the presence of uncertainty. Applications of statistics and probability to real world problems are also presented.

SUPPLEMENTAL MATERIALS:

Transformation of random variables (Slud)

Computer simluation of random variables (Slud)

The Law of Large Numbers (Boyle)

The Central Limit Theorem (Boyle)

#### Prerequisites

MATH 141 or a C- or better in MATH 131

#### Topics

Data summary and visualization

Sample mean, median, standard deviation
*Sample quantiles, *box-plots
(Scaled) relative-frequency histograms

Probability

Sample space, events, probability axioms
Probabilities as limiting relative frequencies
Counting techniques, equally likely outcomes
Conditional probability, Bayes' Theorem
Independent events
*Probabilities as betting odds

Discrete Random Variables

Distributions of discrete random variables
Probability mass function, distribution function
Expected values, moments
Binomial, hypergeometric, geometric, Poisson distributions
Binomial as limit of hypergeometric distribution
Poisson as limit of binomial distribution
*Poisson process

Continuous Random Variables

Densities: probability as an integral
Cumulative distribution, expectation, moments
Quantiles for continuous rv's
Uniform, exponential, normal distributions
*Gamma function and gamma distribution
*Other continuous distributions
*Transformation of rv's (by smoothly invertible functions): distribution function and density
*Simulation of pseudo-random variables of specified distribution (by applying inverse dist. func. to a uniform pseudo-random variable)

Joint distributions, random sampling

Bivariate rv's, joint (discrete) probability mass functions
*Expectation of function of jointly distributed rv's
*Joint and marginal densities
*Correlation, *covariance
Mutually independent rv's. Mean and variance of sums of independent rv's
*Sums of rv's, laws of expectation
Law of Large Numbers, Central Limit Theorem
Connection between scaled histograms of random samples and probability density functions

Point estimation

Populations, statistics, parameters and sampling distributions
Characteristics of estimators : consistency, accuracy as measured by mean square error, *unbiasedness
Use of Central Limit Theorem to approximate sampling distributions and accuracy of estimators
Method of moments estimator
*Maximum likelihood estimator
*Estimators as population characteristics of the empirical distribution

Confidence intervals

Large sample confidence intervals for means and proportions using Central Limit Theorem
*Small sample confidence intervals for normal populations using Student's t distribution
Confidence interval as decision procedure/hypothesis test

Hypothesis Tests

*Hypothesis testing definitions (Null and alternate hypotheses, Type I and II errors, significance level and power, p-values)
*Tests for means and proportions in large samples, based on the Central Limit Theorem
*Small sample tests for means of normal populations using Student's t distribution
*Exact tests for proportions based on binomial distribution

* = optional