Introduction (1) What is statistics? (2) Probability and probability model Exploratory data analysis (1) Sample statistics (mean, variance, median, mode, quantiles) (2) Histogram and box-and-whisker plot (3) Empirical cumulative distribution function? Random variable and probability distribution (1) Definition of random variable (2) Discrete and continuous random variables (3) Cumulative distribution function and probability density function? Characterizing a random variable (1) Expectation, variance, and quantile (2) Moments and moment generating function (3) Expectation of a function of a random variable ? Univariate distributions (I) Discrete distributions (1) Uniform distribution (2) Bernoulli distribution (3) Binomial distribution (4) Negative binomial distribution (5) Geometric distribution (6) Poisson distribution ? Univariate distributions (II) Continuous distribution (1) Uniform or rectangular distribution (2) Normal distribution (3) Exponential distribution (4) Gamma distribution (5) Pearson Type III distribution (6) Log normal and log Pearson Type III distributions ? Joint distribution and conditional distribution (1) Joint cumulative distribution function and joint probability density function (2) Marginal density function (3) Conditional distribution and conditional density function (4) Covariance and correlation (5) Bivariate normal distribution Sampling and sampling distributions (I) (1) Definition of random sample (2) Definition of a statistic (3) Sample moments (4) The Central Limit Theorem ? Sampling and sampling distributions (II) (1) Student's t distribution (2) Chi-square distribution (3) F distribution (4) Order statistics Statistical inference (I) Parameter point estimation (1) Estimators and estimates (2) Method of moments (3) Maximum likelihood method (4) Properties of estimators ? Hypothesis tests (I) (1) Null and alternative hypothese (2) The Critical Region and Test Statistics (3) The Power Function (4) Types of error? Hypothesis tests (II) One-sample tests on the mean and variance (1) Tests on Mean (2) Tests on Variance (3) Paired t-test ? Hypothesis tests (III) Nonparametric Goodness-of-fit (GOF) tests (1) Goodness-of-fit test (2) Chi-square GOF test (3) Kolmogorov-Smirnov GOF test? Linear Regression (I) (1) The Method of Least Squares (2) Coefficient of determination (3) Estimating the variance of Y|x? (4) Confidence intervals of the regression coefficients (5) Hypothesis tests for regression coefficients (6) Confidence interval of the regression estimate (7) Prediction interval of the predicted value

Course Objectives 1. Learn the language and core concepts of probability theory. 2. Understand basic principles of statistical inference (both Bayesian and frequentist). 3. Build a starter statistical toolbox with appreciation for both the utility and limitations of these techniques. 4. Use software and simulation to do statistics (R). 5. Become an informed consumer of statistical information. 6. Prepare for further coursework or on-the-job study. Specific Learning Objectives Students completing the course will be able to: Probability 1. Use basic counting techniques (multiplication rule, combinations, permutations) to compute probability and odds. 2. Use R to run basic simulations of probabilistic scenarios. 3. Compute conditional probabilities directly and using Bayes’ theorem, and check for independence of events. 4. Set up and work with discrete random variables. In particular, understand the Bernoulli, binomial, geometric and Poisson distributions. 5. Work with continuous random variables. In particular, know the properties of uniform, normal and exponential distributions. 6. Know what expectation and variance mean and be able to compute them. 7. Understand the law of large numbers and the central limit theorem. 8. Compute the covariance and correlation between jointly distributed variables. 9. Use available resources (the internet or books) to learn about and use other distributions as they arise. Statistics 1. Create and interpret scatter plots and histograms. 2. Understand the difference between probability and likelihood functions, and find the maximum likelihood estimate for a model parameter. 3. Do Bayesian updating with discrete priors to compute posterior distributions and posterior odds. 4. Do Bayesian updating with continuous priors. 5. Construct estimates and predictions using the posterior distribution. 6. Find credible intervals for parameter estimates. 7. Use null hypothesis significance testing (NHST) to test the significance of results, and understand and compute the p-value for these tests. 8. Use specific significance tests including, z-test t-test (one and two sample), chi-squared test. 9. Find confidence intervals for parameter estimates. 10. Use bootstrapping to estimate confidence intervals. 11. Compute and interpret simple linear regression between two variables. 12. Set up a least squares fit of data to a model.

Walpole, R., et al. (2013). Essentials of Probability & Statistics for Engineers & Scientists, Pearson Education.