Important details
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120 minute take home test.
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Covers everything up to Mar 20: transformations, bivariate random variables, sequences and sampling distributions.
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4 questions. Approximately half applied (working with real problems) and half theoretical (working with mathematical symbols).
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Honour code: No collaboration. No communication about the questions or your answers. Exams should be pledged and signed.
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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
Past tests
Learning objectives
Bivariate 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.
- 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
- 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 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.
- iid
- Chebyshev’s theorem
- The law of large numbers
- The central limit theorem
- Approximately distributed
- 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.