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Lecture 15 on BST 631: Statistical Theory I – Kui Zhang, 10/10/
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Review for Chapter 1 to Chapter 3 Chapter 1 – Probability Theory
set theory – union, intersection, complementation
experiment, sample space, events, operations (union, intersection, complementary), disjoint events
commutatitivity, associativity, distribution law, DeMorgan’s law
probability (definition, properties)
how to compute probabilities^ •
from union, intersection and complements of events
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under equally likely outcomes using counting techniques
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how to count: order, replacement
conditional probability and independence of events, Bayes’ Rule
random variable – definition; continuous versus discrete
cdf, pdf and pmf – definitions,^ •
conditions needed to satisfy each of these functions
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how to use these functions to compute probabilities
Chapter 2 – Transformations and Expectations
functions of random variables and how to derive their distributions^ •
using cdf
Lecture 15 on BST 631: Statistical Theory I – Kui Zhang, 10/10/
2
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Theorem 2.1.5 – monotone function on the whole support of
X
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Theorem 2.1.8 – nonmonotone function but monotone on an appropriate partition of the support of
X
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Theorem 2.1.10 - Probability integral transformation
expected values of functions of random variables
n^ th moment and
n
th central moment
mean, variance and standard deviation of a random variable
X
moment generating functions^ •
make sure to include specific values of t where the mgf exists
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how to use mgf to find the moments
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The application of mgf - the convergence of mgf implies the convergence of random variables
Leibnitz’s rule
interchanging the order of differentiation, integration, summation, limit
Chapter 3 – Common Families of Distributions
Common Discrete Distributions^ •
discrete uniform, Bernoulli, Binomial, hypergeometric, Poisson, geometric, negative binomial
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assumptions behind each of these models
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should know pmf, cdf, support, mean, variance, mgf, special properties if any
Common Continuous Distributions^ •
continuous uniform, gamma, exponential, chi-squared, Weibull, normal, beta, Cauchy, lognormal
