STATISTICS

Course Credits: 3 Units

Prerequisites: Stat 121

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STAT 122 - Probability Theory II

Course Description

This course covers joint, marginal, and conditional distributions; independence of random variables; distributions and expectations of functions of random variables; characterization of F, t, and χ^2 distributions; normal approximation to discrete distributions.

Course Learning Outcomes

After the course, the learner should be able to:

  1. Identify and distinguish between the joint and marginal distributions;
  2. Evaluate joint, marginal and conditional distributions;
  3. Illustrate independence of several random variables through the joint distribution, marginal distribution, expectations and moment generating functions;
  4. Evaluate expectations of several random variables;
  5. Evaluate conditional expectations;
  6. Obtain joint moment generating functions;
  7. State the probability density function of the bivariate normal distribution;
  8. Derive the marginal and conditional densities of the bivariate normal distribution;
  9. Determine the distribution of functions of several random variables using the cumulative distribution function, the moment generating function, and the transformation technique;
  10. Derive the probability density functions of F, t, and χ^2 distributions; and,
  11. Use the normal distribution to approximate the Poisson and Binomial distributions;
Course Outline

UNIT 1. Joint and Marginal Distributions

  1. Joint Distributions
    1. Definition of a Joint Distribution Function
    2. Properties of Joint Distribution Functions
    3. Classification of Joint Distribution Functions
  2. Marginal Distributions
    1. Discrete Case
    2. (Absolutely) Continuous Case

UNIT 2. Conditional Distributions and Stochastic Independence

  1. Conditional Distributions
    1. Discrete Case
    2. (Absolutely) Continuous Case
  2. Independence of Random Variables

UNIT 3. Expectations of Several Random Variables

  1. Expectations of Functions of Several Random Variables
    1. Definition
    2. Properties of Expectation
    3. Some Special Expectations
    4. Inequalities
  2. Conditional Expectation
    1. Definition
    2. Expectation by Conditioning
  3. Joint Moment Generating Functions and Moments
    1. Definition
    2. Generation of Moments
    3. Some Important Results
  4. Bivariate Normal Distribution
    1. Density Function
    2. Moment Generating Function and Moments
    3. Marginal and Conditional Densities

UNIT 4. Distribution of Functions of Random Variables

  1. Distribution of a Function of a Single Variable
    1. Discrete Case
    2. (Absolutely) Continuous Case
  2. Distribution of a Function of Several Random Variables
    1. CDF Technique
    2. MGF Technique
    3. Transformation Technique

UNIT 5. Sampling and Sampling Distributions

  1. Sampling
    1. Populations and Sample
    2. Statistics and Sample Moments
  2. Sample Mean
    1. Sample Mean and Sample Variance
    2. Law of Large Numbers
    3. Central Limit Theorem
  3. Sampling from the Normal Distribution
    1. Sampling Distribution of the Mean when Sampling from a Normal Population
    2. The Chi-square Distribution
    3. The F-Distribution
    4. Student's t-Distribution
  4. Order Statistics
    1. Definition
    2. Distribution of the Minimum Order Statistic
    3. Distribution of the Maximum Order Statistic

UNIT 6.Normal Approximation to Discrete Distributions

  1. Poisson by Normal
  2. De Moivre-Laplace Limit Theorem (Binomial by Normal)