Important details

120 minute take home test.

Covers everything up to Feb 9: probability, discrete random variables, continuous random variables.

4 questions. Approximately half applied (working with real problems) and half theoretical (working with mathematical symbols).

Honour code: No collaboration. No communication about the questions or your answers. Exams should be pledged and signed.

Only outside resources allowed are: a onepage doublesided note sheet and wolfram alpha.
Timeline
 Feb 9: last topics for test
 Feb 14: in class review
 Feb 15: test available
 Feb 16: homework 5 due
 Feb 23: test due at 4pm
Past tests
Learning objectives
Probability
Vocabulary
You should know the definitions of the following words/phrases. (You won’t need to regurgitate these definitions in an exam, but you’ll need them to understand the questions and solve the problems). Where possible, you should be able to express the phrase in both words and mathematics.
 Random experiment
 Sample space
 Event
 Mutually exclusive
 Exhaustive
 Partition
 Probability function, including the three defining criteria, and the other important properties
 Equally likely events
 Combination
 Permutation
 Conditional probability, including the properties that it satisfies
 Independence
 Prior and posterior probability

Basic set algebra.

Given a word problem, define appropriate sample space and events, and convert problem to a mathematical question.

Multiplication principle
 Sampling with and without replacement
 Correctly reduce the number of outcomes if order doesn’t matter

The basic tools of rearranging probability problems (including when it is appropriate to use each tool):
 Taking complements.
 Multiplication rule / replacing intersections with conditioning
 Convert unions to sums
 Convert intersection to products (if independent)
 Law of total probability.
 Bayes rule = multiplication rule + law of total probability

Convert conditional probabilities to natural frequencies.
Random variables
Vocabulary
You should know the definitions of the following words/phrases. (You won’t need to regurgitate these definitions in an exam, but you’ll need them to understand the questions and solve the problems). Where possible, you should be able to express the phrase in both words and mathematics.
 Random variable. You should be able to compare a random variable to a random experiment, and give the difference between a continuous and discrete random variable.
 Probability mass/density function, including the two defining criteria. You should be able to confirm whether or not a function is a pmf/pdf.
 Expectation. Definition and properties of linear operator.
 Mean and variance (including two ways of computing).
 Give the definition of moments and central moments and succinctly describe the difference
 The moment generating function, including the general formula, and how to use it to compute moments.
 Be able to use the moment generating function to compute the mean and variance of a distribution.
 Calculate the mean and variance directly from the pmf/pdf.
 Calculate any moment given the mgf
 Find the mgf for a given pmf/pdf.
 Given a random variable with a known pdf/cdf, and a transformation of that random variable, find the pdf/cdf of the transformed variable (using the distribution technique or the change of variable technique).
 Recognise the special relationships between a random variable X, it’s cdf and the uniform distribution.
 Derive a cdf from a pdf and vice versa.
 Use a table of cdf values to determine the probability that a random variable lies in a given interval.
Distributions
You should be familiar with the definition, sample space, parameters, assumptions, pmf/pdf, mgf, mean and variance of the following distributions:
 Discrete uniform
 Bernoulli
 Binomial
 Poisson
 Uniform
 Exponential
 Gamma
 Normal
You should also be able to:
 Identify the corresponding named distribution and parameters given a pdf or mgf (or assert that it is not from distribution that you know about)
 Select the distribution that best matches the constraint of a real problem, and justify that choice.