stat310

# Test 1

## 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 one-page double-sided 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

## 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

#### Mathematical tools

• 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.

#### Mathematical tools

• 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.