stat310

# Final

• Due May 2, 5pm

• 180 minutes (three hours)

• 8 questions. 3-4 on topics related to inference. Others on everything else

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

• You may use three double-sided pages of notes

## Learning objectives

### 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
• Define parameter space, estimator and point estimate
• Describe the likelihood, and how it differs from the joint pdf
• Motivate why we are interested in inference
• Discuss what criteria we might use to choose between different estimators.
• Identify the random experiment in an inferential setting
• Translate between the criminal justice system and the statistical justice system

### 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.

• Give the distribution of the standard deviation of a sequence of iid normal random variables.

• Describe the distribution of the standard deviation of a sequence of iid normal rv’s.

• Use method of moments to find an estimator for a parameter

• Use maximum likelihood to find an estimator for a parameter

• Recall the five ways to connect an estimator with the true value and a distribution

• Find a confidence interval for a mean, variance or simple function thereof.

• Perform a hypothesis test.

### Chi-square distribution

• Recall the mean and variance
• Show how the normal and chi-squared distributions are related
• Give the distribution of sums and differences of chi-square variables.