#### 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