stat310

Test 2

Important details

• 120 minute take home test.

• Covers everything up to Mar 20: transformations, bivariate random variables, sequences and sampling distributions.

• 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

• Mar 20: in class review
• Mar 22: test available
• Mar 29: test due at 4pm

Learning objectives

Bivariate random variables

Vocabulary

You should know the definitions of the following words/phrases. (You wonâ€™t need to regurgitate these definitions in an exam, but youâ€™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.

• Give the conditions a bivariate pmf/pdf must satisfy.
• The bivariate cdf (and describe why it isn’t as useful as the univariate cdf)
• Joint, conditional and marginal distributions
• Independence
• Covariance and correlation

Mathematical tools

• Compute a probability given a joint pdf/pmf
• Integrate a joint distribution to produce a marginal distribution.
• Given the marginal distributions of two independent random variables, find the joint distribution.
• Given a marginal and conditional distribution, compute the joint distribution.
• Compute a conditional distribution given a joint and a marginal
• Given a joint distribution, determine if the two variables are independent.
• Recognise when the expectation of a product is the product of the expectation.
• Recall the expectation of a sum is always the sum of the expectations.
• Compute covariance and correlation from a joint distribution.
• Give two ways to compute the covariance
• Follow the steps of bivariate change of variables to perform a simple two-d change of variables.

Sampling distribution summary

Vocabulary

You should know the definitions of the following words/phrases. (You wonâ€™t need to regurgitate these definitions in an exam, but youâ€™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.

• iid
• Chebyshev’s theorem
• The law of large numbers
• The central limit theorem
• Approximately distributed

Mathematical tools

• Give the mgf of a sum of independent random variables
• Give the mgf of a sum of iid random variables
• Compare and contast the law of large numbers to the central limit theorem.
• Using the mgf, show that the mean of a sequence of iid normal rv’s is normally distributed
• Using the clt, show the mean of a sequence of any iid rv’s is approximately normally distributed.