200A Final Review
Lec 1&2:
 Be familiar with set operations and properties
 Know the axioms of probability and some properties of probability function
 Understand the definition of conditional probability
 Know the definition of independent events; for given events, be able to tell whether or not they
are independent
 Know, be able to prove and use the Bayes’ rule
 Understand the relationship between disjoint and independent
 Know the difference between mutual independence and pairwise independence
 Understand the fact that a r.v. is a function from the sample space to the real line
Lec 3 & 4
 Know the difference between continuous and discrete random variables
 Be familiar with the properties of cdf, pdf/pmf, such as the sum of pmf over support is 1.
 Ignore the proof for “the sum of negative binomial pmf is 1”
 Know and be able to prove the memoryless property of the exponential and geometric
distributions
 Know how to find the distribution of a function of a random variable
Lec 5&6
 Know how to find the distribution of a function of a random variable
 Know how to calculate expectations and variance. Be familiar with the trick of “kernel of
pdf/pmf”.
 Be able to use common mathematical results such as the Binomial Theorem and Taylor series in
calculations.
 Know how to calculate the expectation of a function of a random variable. Jensen’s inequality
can be skipped.
 Be able to prove and use Thm 2.2.5
 Know the definition of mgf. Be able to use mgf to find moments.
 Know how to identify the distribution of a random variable by looking at its mgf
 Be able to find the mgf of a linear function of independent random variables
 Know how to use mgf to show convergence in distribution
Lec 7:
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The first definition of Poisson Process can be ignored
The equivalence between the two definitions can be ignored
Know the second definition of Poisson Process
Know the connection between waiting time and Poisson process
Be able to calculate conditional distributions
Lec 8-10:
 Everything except location and scale families
Lec 11:
 Know, be able to prove and use iterated expectation
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Be able to use conditional variance identify
Non-central chi-square can be ignored
Lec 12:
 Be able to use several tricks, such as “proportional to”, “kernel of pdf/pmf”, to simplify
calculations for posterior distributions/means/variances
 Know how to find the distribution of a function of several random variables
 Know the definition of covariance and correlations. Be familiar with their properties and be able
to use them
Lec 13&14:
 Understand the relationship between zero covariance/correlation and independence
 Be able to derive covariance from conditional and marginal distributions from multinomial
distribution
 Understand that similar tricks used in the multinomial can be used for Dirichlet
 You don’t have to memorize all the definitions of MVN but you should be familiar with and be
able to prove properties of MVN
Lec 15:
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Know the formula of conditional distribution
Be able to prove the Lemma on page 62
Be able to prove 1,2,4 of the example on page 62
Know how to derive the distribution of the larges and smallest order statistics
Lec 16:
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Be familiar with the definition of convergence in probability
Be able to use the definition to show convergence in probability
Be able to prove and use Chebyshev’s
Be able to do all the examples by yourself
Lec 17&18:
 Know and be able to use the continuous mapping theorem. The proof is not required.
 Understand that there is something called SLLN, which is stronger than WLLN.
 Be familiar with the definition and convergence in distribution.
 Be able to use the definition to show convergence in distribution
 Know the relationship between convergence in probability and in distribution. The proofs are
not required
 Be able use mgf to show convergence in distribution
 Be able to state and prove the CLT
 Be familiar with Slutsky’s theorem. Be able to replicate the proof in the example on page 9
(“convergence 9”)
 Be able to use the first order Delta method to find asymptotic distributions
 Understand and know how to use the result of multivariate CLT
 Be able to use multivariate delta method to solve problems