1 Probability Review

This course builds on topics that you covered in MATH/STAT 354: Probability. Some of you might have taken Probability last semester, and for others it might have been awhile. I certainly don’t expect you to remember everything you learned in that class, but there are some concepts from Probability that we will use fairly often in this class (see the list below): if you’re rusty on any of these, I recommend spending some time reviewing.

Learning Goals

  • Distinguish between important probability models (e.g., Normal, Binomial)
  • Derive the expected value and variance of a single random variable or a sum of random variables
  • Find the distribution of order statistics (e.g., minimum, maximum)
  • Define the moment generating function and use it to find moments or identify pdfs

(Optional) Textbook Reading Guide

Read: Chapters 2–4 (pages 15–277)

Definitions:

  • probability density function (discrete, continuous)
  • cumulative distribution function (discrete, continuous)
  • joint probability density function
  • conditional probability density function
  • independence
  • random variable
  • expected value
  • variance
  • \(r^{th}\) moment
  • covariance
  • random sample
  • order statistic
  • moment generating function
  • Binomial distribution
  • Poisson distribution
  • Geometric distribution
  • Negative Binomial distribution
  • Normal/Gaussian distribution
  • Gamma distribution
  • Exponential distribution
  • Uniform distribution

Theorems:

  • Law of Total Probability – Theorem 2.4.1
  • Bayes’ Theorem – Theorem 2.4.2
  • Relationship between pdf and cdf – Theorem 3.4.1 and Theorem 3.7.3
  • Expected value and variance of linear transformations of random variables – Corollary 3.5.1, Theorem 3.6.2, Theorem 3.9.2, Theorem 3.9.5
  • Relationship between mean and variance – Theorem 3.6.1
  • Finding a marginal pdf from a joint pdf – Theorem 3.7.1 and Theorem 3.7.2
  • Independence of random variables and joint pdfs – Theorem 3.7.4
  • Expected value of a product of independent random variables – Theorem 3.9.3
  • Covariance of independent random variables – Theorem 3.9.4
  • Finding the pdf and cdf of an order statistic – Theorem 3.10.1
  • Using MGFs to find moments – Theorem 3.12.1
  • Using MGFs to identify pdfs – Theorem 3.12.2 and Theorem 3.12.3
  • Central Limit Theorem – Theorem 4.3.2

Note: page and example numbers correspond to the 6th Edition of Larsen & Marx and may not correspond directly to earlier editions of the textbook.