Course slides

WEEK 1

• Before we start - Logistical Remarks
• Why Probability (and Statistics) - An Introduction
• Sample spaces and events (MR 2.1)
• Counting techniques
• The concept of Probability (MR 2.2)
• Axioms of Probability
• Algebraic rules for probabilities
• Addition rules (MR 2.3)
• Conditional Probability (MR 2.4)
• Multiplication and total probability rules (MR 2.5)
• Independence (MR 2.6)
• Bayes' Theorem (MR 2.7)
• Examples and Discussion
• Simpson's Paradox
• Introduction to Random Variables (MR 2.8)

WEEK 2

• Definition of Random Variables (MR 2.8)
• Continuous vs. Discrete
• Discrete Random Variables (MR 3.1)
• Probability Mass Function (MR 3.2)
• Cumulative Distribution Function (MR 3.3)
• Mean and Variance (MR 3.4)
• Functions of random variables
• Independence of Random Variables (MR 5-1.4)
• Mean and Variance of sums of independent random variables (without proof)
• Specific discrete distributions
• Uniform (MR 3.5)
• Bernoulli (MR 3.6)
• Bernoulli trials
• Binomial (MR 3.6)
• Tables and Computational Tools for the evaluation of event probabilities.
• Geometric (MR 3.7)
• Memoryless property

WEEK 3

• Specific discrete distributions
• Negative Binomial (MR 3.7)
• Hypergeometric Distribution (MR 3.8)
• The Poisson Distribution (MR 3.9)
• The relation between Binomial and Poisson distributions
• Introduction to Continuous Random Variables
• Continuous Random Variables (MR 4.1)
• Probability Density Function (MR 4.2)
• Some Important Properties
• Cumulative Distribution Function (MR 4.3)
• Mean and Variance of Continuous Random Variables (MR 4.4)
• Many Examples
• Specific Continuous Distributions
• Uniform Distribution (MR 4.5)
• Normal Distribution (MR 4.6)
• Motivation and some properties
• The standard normal distribution
• Basic Properties

• The standard normal distribution
• Basic Properties
• Standardizing a Normal random variable
• Using probability tables and computational tools
• The Exponential Distribution (MR 4.8)
• The Poisson Process (not clearly explained in the book)
• Specific Continuous Distributions
• The Poisson Process (not clearly explained in the book)
• Erlang Distribution (MR 4.9)

WEEK 4

• Weibull Distribution (MR 4.10)
• Joint Probability Distributions (discrete case) (MR 5.1)
• Joint Probability Mass Functions
• Marginal p.m.f.'s
• Conditional p.m.f.'s
• Independence of random variables (MR 5-1.4 and 5-1.5)
• Examples
• Continuous random variables (only basic ideas)
• Means, Variances, Covariances and Correlation (MR 5-2)
• Correlation and independence are not the same thing - example
• Linear Functions of normal random variables (MR 5-4)
• The Law of Large Numbers (not stated in the book)

• The Central Limit Theorem (stated only in page 239 of MR)
• The normal distribution as an approximation of (MR 4-7)
• Binomial distribution
• Poisson distribution
• Introduction to Simulation and Monte Carlo methods
• Historical Notes
• Pseudo-Random Sequences
• Simulating Random Variables
• Discrete Random Variable
• Generating Bernoulli Random Variables
• Generating Binomial, Geometric and Negative Binomial Random Variables
• Generating arbitrary discrete random variables
• Simulating Random Variables
• Continuous Random Variables
• Inverse Transform Method

WEEK 5

• Simulating Random Variables
• Continuous Random Variables
• Rejection Method
• Addressing Complicated Problems with Simulation
• Estimating Probabilities
• Approximate analysis using the CLT
• Conservative results using Chernoff's bound
• Estimating Means and Variances
• Approximate analysis using the CLT
• What is Statistics?
• Descriptive Statistics: Numerical vs. Graphical Representations (MR Chapter 6)
• Numerical Summaries of Data
• The Histogram and Frequency Distributions

WEEK 6

• Descriptive Statistics: Numerical vs. Graphical Representations (MR Chapter 6)
• Numerical Summaries of Data
• The Histogram and Frequency Distributions
• The Box-Plot
• Time-Sequence Plots
• Quantile-Quantile Plots

• Introduction to Point Estimation (MR 7.1)
• The Notion of a Random Sample (MR 7.2)
• Estimators and Estimates (MR 7.2)
• Criteria for the Evaluation of Estimators (MR 7.3)
• Mean Squared Error (MSE)
• Bias, Variance, and Standard Error
• Bias-Variance decomposition

• Methods of Point Estimation (MR 7.4)
• Method of Moments (MR 7.4.1)
• Maximum Likelihood Estimation (MR 7.4.2)

• Recap of the Method of Moments and Maximum Likelihood Estimation
• Important Examples
• Beyond the random sample case - an example
• Introduction to Interval Estimators and Confidence Intervals (MR 8.1)

WEEK 7

• Confidence Intervals for the mean when variance in known (MR 8.1)
• One and two-sided intervals
• Choice of sample size
• Approximate Confidence Intervals for large samples (MR 8-1.5)
• Confidence Intervals for the mean when the variance is unknown (MR 8.2)
• Confidence Intervals for the variance and standard deviation (MR 8.3)
• Confidence Intervals for proportions (MR 8.4)
• Prediction Intervals (MR 8-7.1)
• Introduction to Hypothesis Testing (MR 9.1)
• Statistical Hypothesis
• Tests and Tests Statistics
• Type I and Type II errors

• Introduction to Hypotheses Testing (MR 9.1)
• P-Values
• Some Useful Tests
• Tests for the mean, known variance (MR 9.2)
• Choice of Sample Size (MR 9-2.2)
• Large Sample Tests (MR 9-2.3)
• Tests for the mean, unknown variance (MR 9.3)

WEEK8

• Tests for the variance (MR 9.4)
• Tests for the proportions
• Goodness-of-Fit (GoF) tests for normality
• Introduction to Linear Regression (MR 11.1)
• Simple Linear Regression (MR 11.2)
• Properties of Least Squares Estimators

• Properties of Least Squares Estimators
• Hypothesis Testing for Linear Regression Models
• Approach based on t-tests
• ANOVA approach
• Confidence Intervals for the parameters in Linear Regression Models
• Confidence Intervals for the Mean Response
• Prediction of New Observations
• Adequacy of the Regression Model
• Residual Analysis
• Coefficient of Determination
• Final Remarks

WEEK 9

• Solving exercises from previous exams and similar activities
• exam_July2_2014.pdf
• power.pdf
• dataset_quick_assignment.txt ; R_example.R