Math 425-010 - Introduction to Probability

Fall 2019

Course Info

Instructor: Corey Everlove

Email: everlove@umich.edu

Office: East Hall 4860

Final Exam - Thursday Dec 19, 10:30-12:30, Weiser 269

Coverage notes

Suggested study problems

Bring: 3×5 card with notes, calculator.

Office hours before the final exam: 2:00-4:00 on the following days:

Class Schedule

Date Sections(s) covered Topics Homework (from Ross, 9th Edition) Notes
Thurs, Dec 19 (final exam) Final exam and solutions
Tues, Dec 10 8.3
  • More examples with the central limit theorem
  • Review
 We also talked about the St. Petersburg paradox.
Thurs, Dec 5 8.3, 8.4
  • Strong law of large numbers
  • Central limit theorem
Tues, Dec 3 7.3, 7.4
  • Computing variance of a random variable X by writing X as a sum of indicator RVs
  • Covariance of two random variables
 In Section 7.3, we are only covering the parts about computing the second moment (the expectation of X²), not the higher moments.
Tues, Nov 26 5.6.1, 5.6.4, 6.3.2, 6.5(page 255)
  • Poisson processes and the gamma distribution
  • The beta distribution
 The material covered today on gamma and beta distributions will not be on the final exam.
Thurs, Nov 21 more 7.2, 7.5
  • More examples of computing probabilities and expectations using linearity of expectation and conditioning
 HW 10 due Thurs Dec 6:
problems 7.23, 7.48, 7.49, 7.53, 7.56, 7.58
solutions		 Our discussion of the "best prize problem" at the end of class was a little rushed. See Example 5k on page 326 for more details.
Tues, Nov 19 7.2, 7.5
  • More examples of computing expectations using linearity of expectation
  • Computing expectations by conditioning
 Our final exam is during the regularly scheduled time: Thursday, December 19, 10:30-12:30.
Thurs, Nov 14 (exam) HW 9 due next Thurs Nov 21:
problems: 7.2, 7.8, 7.10, 7.12, 7.13
solutions		 Second midterm exam and solutions
Stats: median=38.5, mean=38.4
Tues, Nov 12 7.1, 7.2
  • Many examples using linearity of expectation
 Exam Thursday in class - we will try to start right at 8:30.
Thurs, Nov 7 6.5, review
  • More on conditional distributions
  • Review problems
 No homework due next week because of exam.
Good problems to try:
6.28, 6.31 (see example 3c), 6.40, 6.41
Self-Test problems from Ch 4, 5, and 6.		 We also discussed Buffon's needle problem.
Tues, Nov 5 6.3, 6.4, 6.5
  • Sums of independent continuous RVs - PDF of X+Y is the convolution of the PDFs of X and Y
  • Sums of independent normal RVs are normal
  • Conditional distributions (discrete and continuous cases), including the conditional PDF of X given Y=y
 Topics covered on the second midterm exam
Thurs, Oct 31 6.1, 6.2
  • More examples with jointly distributed RVs
  • Independence of RVs
  • Characterization of the normal distribution (as in Example 2e on p232)
 HW 8 due next Thurs Nov 7:
problems: 5.39, 6.2, 6.9, 6.13, 6.27, 6.55
solutions
Tues, Oct 29 6.1 (and a note on 5.7)
  • Jointly distributed random variables (both discrete and continuous)
  • Joint PDFs and CDFs, marginal PDFs and CDFs
 The second midterm exam will be Thursday, November 14 in class. Please let me know if you have a previously scheduled commitment at that time.
Thurs, Oct 24 5.4, 5.5
  • More on normal distributions and the standard normal CDF
  • Normal approximation to the binomial distribution (DeMoivre-Laplace Theorem)
  • Exponential distribution is memoryless
 HW 7 due next Thurs Oct 31:
problems 5.13, 5.16, 5.20, 5.22, 5.23, 5.34
Note: in 5.20, 5.22, and 5.23, when it asks for approximation, you should use the normal distribution to approximate the binomial distribution.
Solutions		 We also talked a bit about the Two Envelopes Problem
Tues, Oct 22 5.3, 5.4
  • Reviewed some properties of PDFs and CDFs, including interpreting values of a PDF
  • Uniform RVs
  • Normal RVs (computed the integral, expectation, and variance of the standard normal and derived a formula for the PDF of the normal with mean μ and variance σ²)
Thurs, Oct 17 4.7, 5.1-5.2
  • Poisson RVs and the Poisson approximation to the binomial distribution
  • Continuous RVs (density functions, distribution functions, and expected value)
 HW 6 due next Thurs Oct 24:
problems 4.55, 4.59, 4.60, 4.65, 5.4, 5.6
Note: In 4.59 and 4.65, when it asks for approximate probabilities, you should use the Poisson approximation.
Solutions
Thurs, Oct 10 4.8-4.9, some of 4.7
  • Geometric RVs, negative binomial RVs
  • Motivation for Poisson RVs
 HW 5 due next Thurs Oct 17:
Ch. 4, problems 4.20, 4.71, 4.72, 4.74
(short assignment due to Fall Break)
Solutions		 First midterm exam and solutions
Stats: median=48, mean=47.5
Tues, Oct 8 4.1-4.6
  • Review of random variables, expectation
  • Variance
  • Bernoulli RVs and binomial RVs
Sep 2-Oct 3 1.1-4.4 See Canvas for a detailed schedule and the assigned homeworks 1-4.