Mathematics 375 – Probability Theory
Review Sheet, Final Exam
December 1, 2011
General Information
The final examination for this course will be given at 3:00 pm on Friday, December 16
in our regular class room, Swords 359. The exam will be roughly twice the length of one of
the two midterms and I will let you continue working on it as long as you want (even after
the formal end of the exam period at 5:30pm, if necessary). As was true on the midterms,
I will let you bring a sheet of information for your use on the exam – for this exam, you
can place anything you want on one side of an 8.5 ×11 inch piece of paper. I will provide
copies of whatever tables from the text might be needed to work some of the problems.
Topics to be Covered
This will be a comprehensive exam – all topics we have discussed this semester are
“fair game.” This includes:
1) Descriptive statistics such as the mean, standard deviation, frequency histograms, etc.
The “empirical rule”.
2) Discrete sample spaces and counting techniques (especially in connection with the
“sample point method” for the probability of an event): the m×nrule, permutations,
binomial and multinomial coefficients.
3) The “event composition method” for probabilities
4) Conditional probabilities, independence of events, Bayes’ Rule, the Law of Total Prob-
ability
5) Discrete random variables: probability distribution functions, cumulative distribution
functions, expected values and variances of functions of a discrete random variable,
moment generating functions. Know the situations leading to binomial, geometric,
Poisson, and hypergeometric random variables and how to apply them.
6) Continuous random variables: probability distribution functions, cumulative distri-
bution functions, expected values and variances of functions of a continuous random
variable, moment generating functions. Know the situations leading to uniform, ex-
ponential, gamma, beta, and normal random variables and how to apply them.
7) Tchebysheff’s Theorem.
8) Multivariate probability distributions: joint densities, marginal and conditional den-
sities, expected values in this setting, conditions for independence, the covariance and
the general formula for the variance of a linear combination of random variables.
9) Using moment generating functions and the uniqueness theorem to determine the
distribution of a random variable.
10) The method of distribution functions to determine the distribution of a random vari-
able.
11) Know the statement and proof of the Central Limit Theorem.
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