MA396 In-Class Final Skill Inventory
The in-class portion of the final exam will emphasize the mathematical
theory underlying statistics.
The test will be open book, open notes.
In the material that follows, sections in the Larson and Marx text are given for reference.
In some cases online materials are also referenced.
Basic Probability Theory
You should be familiar with the following topics:
- Continuous and discrete univariate probability density functions (Sections 3.3, 3.4)
- Be able to determine whether a given function is a valid probability density
- Be able to find the cumulative distribution function given the density function
- Univariate cumulative distribution functions (Section 3.4)
- Be able to determine the probability density function given the cumulative distribution function.
- Joint (multivariate) probability density functions (Section 3.7)
- Be able to determine whether a function is a valid multivariate pdf
- Given a multivariate probability density, find the marginal pdfs
- Given a multivariate probability density function, find the conditional pdfs
Expectation and Moments
You should be familiar with the following topics:
- Expected value (Section 3.5)
- Given the pdf or cdf of a random variable x, find the expected value E(x) directly from the definition
- Be able to find raw moments and central moments from the definition
- Be familiar with algebraic manipulation of expectations
- Given the vector of expected values and the variance-covariance matrix of a vector of random
variables, be able to find the expected value and variance-covariance matrix of a linear combination of the original variables
- Moment generating functions
- Be able to find a moment generating function from the pdf
- Be able to use the moment generating function to find moments
- Be able to find the moment generating function of a linear combination of random variables
- Be able to use the moment generating function to determine the type of probability distribution of a linear combination
- Given a multivariate moment generating function, be able to find the moment generating functions of the maginals
Estimation
You should be familiar with the following topics:
- Maximum likelihood estimates
- Given a pdf, be able to construct the likelihood function of a random sample from that population
- Method of moments estimates
- Given a density function, find method of moments estimates using a random sample
- Properties of estimators: Be familiar with the following definitions:
- Interval estimate
- Unbiased estimator
- Efficient estimator and the Cramer-Rao lower bound
- Sufficient estimators
- Consistency
Hypothesis Testing
You should be familiar with the following terms:
- Null hypothesis
- Alternative hypothesis
- Two-sided alternative
- One-sided alternative
- Type I error
- Type II error
- Power of a test
Distributions
You should be familiar with the following terms:
- The Central Limit Theorem
- The chi square distribution
- The t distribution
- The F distribution