MATH 340, Study Guide for the Final Exam

5/12/2013

Test coverage

1. Combinatorial Analysis
   1. The basic principle of counting (1.2)
   2. Permutations (1.3)
   3. Combinations (1.4)
   4. Multinomial coefficients (1.4)
2. Axioms of Probability
   1. Sample space and events (2.2)
   2. Axioms of probability (2.3)
   3. Properties of probability (5.4)
   4. Classical definition of probability (sample spaces with equally likely outcomes) (2.5)
3. Conditional probability and independence
   1. Conditional probabilities (3.2)
   2. Bayes' formula (3.3)
   3. Independent events (3.4)
4. Random variables
   1. Random variables: examples (4.1)
   2. Discrete random variables: probability mass functions (4.2)
   3. Expected value (4.3)
   4. Expectation of a function of a random variable (4.4)
   5. Variance (4.5)
   6. Bernoulli and binomial random variables (4.6)
   7. The Poisson random variable 4.8
5. Continuous Random Variables
   1. Introduction: concept of probability density (5.1)
   2. Expectation and Variance (5.2)
   3. The uniform random variable (5.3)
   4. Normal random variables (5.4)
   5. The normal approximation of the binomial distribution (5.4.1)
   6. Exponential random variables (5.5)
   7. Skip: 5.5.1 to end of the chapter, except 5.6.1
6. Jointly distributed random variables
   1. Joint distribution functions (6.1)
   2. Independent random variables (6.2)
   3. Sums of independent random variables (6.3)
   4. Skip: 6.4 to end of the chapter
7. Properties of expectation
   1. Introduction (7.1)
   2. Expectations of sums (7.2)
   3. Covariance, variance of sums (7.4)
   4. Moment generation functions (7.7)
8. Limit theorems
   1. Weak law of large numbers. Chebyshev's and Markov's inequalities (8.2)
   2. The central limit theorem. Approximation of sample means by normal random variables (8.3)

Key concepts

• Basic principle of counting, permutations, combinations, binomial theorem
• Sample space, outcomes, events. Probability as a function of event
• Classical definition of probability
• Conditional probability, independence
• Bayes' formula
• Discrete random variables, probability mass functions, cumulative distribution functions
• Expectation, variance in the discrete case
• Formula for the expectation of a function (w/o proof)
• Continous random variables, probability densities, cumulative distribution functions
• Expectation, variance; expectation of a function of a random variable
• Continuous probability distributions: uniform, normal, exponential, gamma
• Discrete probability distributions: Bernoulli, binomial, Poisson
• The DeMoivre-Laplace limit theorem and its use
• Joint distribution functions, densities. Marginal densities
• Independence of random variables. Equivalent definitions: product rules for probabilities, density functions and CDFs
• Convolution of density functions and the way to compute density for the sum
• Computing density for other combinations of random variables (XY, X/Y,...) using CDF
• Expectations, variances applied to the case of several random variables
• Covariance, variance of the sum formula
• Moments, moment generating functions
• Chebyshev and Markov inequalities
• Weak law of large numbers
• The central limit theorem

Examples of random variables

• Bernoulli
• Binomial
• Poisson
• Uniform
• Exponential
• Normal
• Gamma

Examples of theoretical questions

• Show computation of expectation or variance for one of the random variables from above list
• Proofs of basic laws of probabilities based on the axioms (cf. formulas for the intersections, inequalities for the unions, Bonferroni's inequality etc.)
• Derive Bayes' formula or a similar rule involving conditional probability
• Derive formulas for expectations and variances of linear combinations involving random variables
• Derive a formula for the density of the sum of two continuous random variables
• Derive a formula for a moment generating function for Bernoulli, Binomial, Poisson, uniform, exponential or normal random variable
• Derive Chebyshev's or Markov's inequality from basic principles
• Formulate weak law of large numbers or the central limit theorem and apply in a specific example