Probability & Statistics

The course begins by building a strong foundation in elementary probability and random variables, key concepts for understanding uncertainty and data analysis. Topics include Bayes' theorem, combinatorics, continuous random variables, and methods for calculating distributions of functions of random variables. By mastering these essential ideas, students are well-prepared to confidently engage with more advanced statistical inference concepts and tackle complex real-world problems.

After studying well-known discrete and continuous random variables, students will delve deeper into the concept of moments. They will learn to calculate expectations, variances, and other fundamental properties of random variables using moments, gaining insight into the behavior and characteristics of distributions. The course also introduces moment-generating functions, exploring their properties and applications. In particular, students will learn how these functions are used to derive the distributions of combinations of random variables, providing a powerful tool for advanced statistical analysis.
In this course, students explore various methods for combining random variables. Key topics include joint, marginal, and conditional distributions and their connections to expectation, variance, and independence within the context of joint distributions.

The second half of the course focuses on statistical inference, covering key topics such as point estimation, the central limit theorem, maximum likelihood estimation, confidence intervals, and hypothesis testing for one-sample and two-sample procedures. Additional topics include one-factor analysis of variance (ANOVA), correlation, regression, and an introduction to chi-square goodness-of-fit tests.

Probability & Random Variables

• Understanding and applying the law of total probability and Bayes' Theorem.

• Calculating permutations and combinations, including cases with repetition, and modeling real-world and mathematical problems using combinatorial techniques.

• Understanding and applying the definition of a probability density function (PDF) for a continuous random variable and calculating probabilities using PDFs.

• Understanding and applying the definition of a cumulative distribution function (CDF) for a continuous random variable and calculating probabilities using CDFs.

• Defining and computing median, quartiles, percentiles, and modes for continuous random variables.

• Understanding how continuity corrections are applied to approximate discrete random variables as continuous.

• Understanding how to simulate data from a given probability distribution.

• Applying the distribution function and change-of-variables methods to transform random variables.

Expectation

• Calculating expected values, variance, and moments for discrete and continuous random variables.

• Understanding and applying properties of expectation and variance.

• Calculating expectations of functions of random variables using LOTUS.

• Calculating moment-generating functions (MGFs) for discrete and continuous random variables.

• Using MGFs to calculate moments and variances or random variables.

• Deriving MGFs for well-known discrete and continuous distributions.

• Deriving and applying properties of MGFs to find MGFs of linear combinations of random variables.

• Applying the uniqueness property of MGFs to derive distributions of sums of random variables with known distributions.

Discrete Probability Distributions

• Constructing and applying the probability mass function (PMF) and cumulative distribution function (CDF) of well-known probability distributions, including:

  • The discrete uniform distribution

  • The Bernoulli distribution

  • The binomial distribution

  • The Poisson distribution

  • The geometric distribution

  • The negative binomial distribution

• Understanding the relationship between the Bernoulli and binomial distributions and the geometric and negative binomial distributions.

• Solving real-world situations using these distributions and understanding the conditions where modeling with a known distribution is appropriate.

• Approximating a binomial random variable using a Poisson distribution.

Continuous Probability Distributions

• Calculating probabilities using the standard normal distribution, including understanding and applying its symmetry properties.

• Modeling using the normal distribution and solving problems via z-scoring and using the CDF of the standard normal distribution, percentage points tables, and the empirical rules.

• Understanding and applying the normal approximations for binomial and Poisson random variables.

• Solving real-world and mathematical problems using the following continuous probability distributions:

  • The continuous uniform distribution

  • The exponential distribution

  • The gamma distribution

  • The chi-square distribution

  • The student's T-distribution

  • Fisher's F-distribution

• Understanding the relationship between the exponential and gamma distributions, the chi-square and normal distributions, the student and normal distributions, and the chi-square and F-distributions.

Combining Random Variables

• Analyzing joint, marginal, and conditional distributions for discrete and continuous random variables.

• Understanding the definition of independence for discrete and continuous random variables and solving related problems.

• Calculating expected values and variances from joint and marginal distributions and exploring conditional expectation and variance.

• Combining multiple random variables and solving related problems, including I.I.D. binomial, Poisson, and normal random variables.

• Deriving PMFs for trinomial and multinomial distributions using combinatorial arguments and solving related real-world and mathematical problems.

• Solving problems using the bivariate normal distribution.

• Applying covariance properties, variance of sums, and correlation coefficients to combinations of random variables.

Parametric Inference

• Understanding the concept of a statistic and sampling distribution, including the sample mean, sample variance, pooled variance, and biased and unbiased estimators.

• Deriving and applying point estimates for expected values and variances of sample means.

• Solving problems involving sample means from normal populations.

• Understanding how elements sampled from finite populations are dependent and use this to solve related problems.

• Deriving and applying point estimates for means and variances of sample proportions.

• Using the method of moments to derive point estimates for samples drawn from one-parameter and two-parameter distributions, and relating this method to the law of large numbers.

• Understanding and applying the central limit theorem (CLT) to approximate distributions of sample means and sums of I.I.D. random variables and solve real-world problems.

• Applying the CLT to approximate distributions of sample proportions and solve real-world problems.

• Applying finite population corrections when estimating distributions of sample means and proportions.

• Constructing likelihood and log-likelihood functions for samples drawn from discrete and continuous populations and using maximum likelihood to estimate population parameters.

Confidence Intervals

• Constructing and interpreting single-sample confidence intervals for:

  • Population means, including cases with unknown population variances and finite population corrections.

  • Population proportions, including cases with unknown population variances and finite population corrections.

  • Population variances drawn from normal distributions.

• Constructing and interpreting two-sample confidence intervals for:

  • Differences in population means, including cases with unknown population variances.

  • Differences in population proportions.

  • Paired samples.

• Estimating appropriate sample sizes for means and proportions.

Hypothesis Testing

• Essential concepts related to hypothesis testing: statistical significance, one-tailed and two-tailed tests, critical regions, critical values, and type I and type II errors.

• Constructing and interpreting single-sample hypothesis tests for:

  • Population proportions, comparing data against a binomial distribution model.

  • Population rates, comparing data against a Poisson distribution model.

  • Population means, including cases with unknown population variances.

  • Population variances drawn from normal distributions.

• Constructing and interpreting two-sample hypothesis tests for:

  • Differences in population means, including cases with unknown population variances.

  • Differences in population proportions.

  • Paired samples.

  • Differences in variances.

• Analyzing grouped data using within-groups and between-groups variation.

• Deriving and calculating SSW, SSB, and SST, understanding the relationship between them and the sample variance.

• Constructing a one-way ANOVA table and testing a hypothesis comparing the equality of multiple means.

Regression

• Computing Pearson's correlation coefficient and fitting linear regression models.

• Computing Spearman's rank correlation coefficient and understanding its relationship with Pearson's coefficient.

• Constructing confidence intervals for linear regression parameters.

• Solving least-squares and regression problems (including polynomial and multiple linear regression) using techniques from linear algebra.

Nonparametric Inference

• Conducting chi-square goodness-of-fit tests for various distributions.

• Testing for independence and homogeneity using chi-square methods.

• Solving problems relating to order statistics.