This course will cover Chapters 1-5 in Leon-Garcia, and should provide a strong foundation in the mathematics of probability. Probability, the study of random or chance processes, is widely used in many fields of study. It was first studied by and for gamblers for their games of chance. Today it is fundamental to modern physics, genetics, computer science, electrical engineering, and a wide spectrum of other areas. It is particularly central to statistics, the study of data collection and analysis, as almost all statistical methods are founded upon probabilistic models.

Prerequisites: You should have a good background in calculus, including multivariable calculus (Math 2210 or equivalent), for this course. It would be helpful to have had Math 4200 or another course with an emphasis on mathematical proofs, but this is not required.

Homeworks will be due in class every Wednesday. The grade for the homework will be reduced by 10% if it is turned in late on Wednesday, and another 10% for every working day it is late after that unless prior permission is given.

The homeworks will primarily consist of problem sets taken from Leon-Garcia. All problems will be worth 10 points and contribute the same amount to your final grade, so the value of a perfect score may vary from week to week. Please make things easy for me and the grader; make your homeworks easy to read and grade. Use one side of the paper, write neatly, and leave plenty of space. We will not grade a paper we can't read. Also, show your work. Full credit will not, in general, be given for just the answer. If your answer is wrong, you will probably receive partial credit if you show your work, but not otherwise. Finally, you may help each other with your homeworks, but I expect what you turn in to be your own work. Helping does not mean simply copying what someone else has put down.

There will be two midterms, on June 30 and July 21, and a final exam on August 4. All tests will be in class at the usual class time, and will consist of problems similar to those on the homeworks.

Your grade in the course will be broken down as follows:
    Homework: 30%
    Midterms: 20% each
    Final: 30%

1. Basic Probability: sample spaces; axioms of probability; counting methods for computing probabilities; permutations and combinations; conditional probabilities; law of total probability; Bayes' rule; independence.

2. Random Variables: discrete random variables; cumulative distribution functions; probability mass functions; bernoulli, binomial, negative binomial, hypergeometric, and Poisson distributions; continuous random variables; probability density functions; uniform, exponential, gamma, normal, and beta distributions; distributions of functions of random variables; expectation, mean, and variance; Markov's and Chebyshev's inequalities; moment-generating, characteristic, and probability-generating functions.

3. Random Vectors: joint, conditional, and marginal distributions, densities, and mass functions; multinomial distributions; bivariate normal distributions; independence of random variables; functions of jointly distributed random variables; covariance and correlation; extreme values and order statistics.

4. Limit Theorems: convergence in probability; the weak law of large numbers; convergence with probability 1; the strong law of large numbers; convergence in distribution; the central limit theorem.

5. Distributions Related to the Normal Distribution: the , t, and F distributions; normal distribution sample mean and variance.

If a student has a disability that will likely require some accomodation by the instructor, the student must contact the instructor and document the disability through the Disability Resource Center, preferably during the first week of the course. Any requests for special considerations relating to attendance, pedagogy, taking of examinations, etc. must be discussed with and approved by the instructor. In cooperation with the Disability Resource Center, course materials can be provided in alternative formats - large print, audio, diskette or Braille.

The above schedule and procedures in this course are subject to change in the event of extenuating circumstances.

• Homework #1 (Due June 14):
  • Chapter 1: 1abd, 3, 5, 7, 10a, 11b

• Homework #2 (Due June 21):
  • Chapter 2: 2, 3, 7, 13, 17, 18, 19, 25 (use ), 26, 33, 34, 39, 40, 42, 43
• Homework #3 (Due June 28):
  • Chapter 2: 48, 49, 51, 53, 58, 59, 62, 64, 72, 77, 78, 80

  • A) Prove Bonferroni's inequality:
    
    
    

  • B) In a game of poker, show that the probabilities that a five-card hand will contain (a) a straight (five cards in numerical order), (b) four of a kind, and (c) a full house (three cards of one rank, two of another) are (a) .0035, (b) .00024, and (c) .0014, respectively.

  • C) Suppose that chips for an integrated circuit are tested and that the probability that they are detected if defective is .95, and the probability that they are declared sound if in fact they are good is .97. If .5% of the chips are faulty, what is the probability that a chip that is declared faulty is actually good?

• Homework #4 (Due July 5):
  • Chapter 3: 2, 4, 7, 9, 12, 17, 19, 21, 35, 37, 38, 42, 44, 46, 47

• Homework #5 (Due July 12):
  • Chapter 3: 52, 53, 56, 59, 60, 63, 65, 68, 70, 80, 81, 88, 90, 93, 100

• Homework #6 (Due July 19):
  • Chapter 3: 153
    
  • Chapter 4: 4, 5, 9, 10, 11, 13, 19, 20, 22, 23, 30, 31, 32, 37

• Homework #7 (Due Friday, July 28):
  • Chapter 4: 39, 49, 50, 51, 54, 55, 59, 62, 64, 65
    Hint for 54: There are two solutions for X1 and X2 in terms of M and V. The p.d.f. of M and V needs to take both into account.
    64, 65: Also calculate the correlation coefficient.

• Homework #8 (Due August 2):
  • Chapter 4: 70, 72, 73, 99, 101
    
  • Chapter 5: 1, 17, 24, 30
    1: That should be W = X + Y + Z, and find the mean and variance of W.
    

  • A) Suppose that the number of insurance claims, N, filed in a year is distributed Poisson with E(N) = 10,000. Use the normal approximation to the Poisson to approximate P(N>10,200).