Midterm 2 study guide

Course

STAT425, Fall 2023

Updated

January 5, 2025

Details and scope

The final exam will be given at the following times:

  • Section 1 Wednesday, December 13, 4:10pm – 7:00pm in Construction Innovations Center C201
  • Section 2 Friday, December 15, 4:10pm – 7:00pm in Construction Innovations Center C201

You are expected to sit for the exam with your scheduled section unless prior arrangements have been made. Logistical details are as follows:

  • 5 questions in length
  • calculators allowed but not required
  • three 8.5 x 11 pages of notes are allowed
  • scratch paper provided
  • table of common distributions provided

The exam is comprehensive but focuses more on the latter portion of the class — random variables, expectations, and joint distributions. You can expect one question pertaining to probability axioms, basic probability rules, conditional probability, or probability on finite sample spaces; the remaining questions will focus on distributions of random variables and random vectors.

Key concepts

The following is a short list of central concepts from the course.

  1. Probability axioms and basic probability calculus
  2. Conditional probability
  3. Independence of events
  4. Bayes’ rule
  5. Probability on finite sample spaces
  6. Random variables
  7. Cumulative distribution function
  8. Probability mass function
  9. Probability density function
  10. Expectation of a function of a random variable
  11. Mean, variance, and moments of a random variable
  12. Moment generating function
  13. Common parametric families (e.g., Bernoulli, binomial, Gaussian, etc.)
  14. Random vectors
  15. Joint PMF/PDFs
  16. Transformations of random vectors
  17. Expectation of a function of a random vector
  18. Covariance
  19. Correlation
  20. Conditional distributions
  21. Independence of random variables

The final involves using these concepts for problem-solving based on definitions, associated properties, and examples developed in class and on homework assignments. Questions assess (a) your understanding of definitions and (b) your ability to apply them in a problem-solving context. Usually, the best starting point for resolving a problem is to identify givens and apply relevant definitions to move towards a solution; some credit is always awarded for answers that interpret problems appropriately. Thus, a good starting point for review is to cement your fluency with definitions of the above concepts and your understanding of how concepts are related.

Preparations

Suggested preparations are:

  1. Review class notes (posted and your own)
  2. Review homeworks, midterms, and posted solutions
  3. Work the practice problems below
  4. Prepare your notesheet(s)

Be sure to also leave some time between your studying and sitting for the exam.

Practice problems

These problems are intended to help you prepare. They are not representative of the length of exam problems, but are closely related to exam questions in some instances.

  1. Let \(X \sim N(\mu, 1)\).

    1. Find the distribution of \(X^2\) when \(\mu = 0\). Is this from a common family?
    2. Find the distribution of \(X^2\) when \(\mu \neq 0\). Is this from a common family?

  2. Suppose \(X, Y\) are each binary random variables with joint distribution given by the table below. Based on the table:

    1. Find the marginal distributions
    2. Determine whether \(X \perp Y\)
    3. Find the conditional distribution of \(X\) given \(Y = 0\)
\(X = 0\) \(X = 1\)
\(Y = 0\) 0.1 0.4
\(Y = 1\) 0.2 0.3

  1. Let \((X, Y)\) be distributed according to the PDF \(f(x, y) = e^{-y}\) for \(0 < x < y < \infty\).

    1. Find the marginal and conditional distributions of \(X, Y, X|Y, Y|X\)
    2. Find the distribution of \(X + Y\)

  2. Let \(X, Y\) be random variables and consider \(U_1 = a_1X + b_1Y + c_1\) and \(U_2 = a_2X + b_2Y + c_2\) for constants \(a_1, a_2, b_1, b_2, c_1, c_2\). Express \(\text{cov}(U_1, U_2)\) in terms of the variances and covariance of \(X, Y\).

  3. Let \(X, Y\) be independent Poisson random variables, each with parameter \(\lambda\). Find the MGF of \(U = aX + bY\).

  4. Let \(X, Y\) be uniformly distributed on the unit circle. Find the PDF and show that the random variables are uncorrelated, but dependent.