STAT 3375Q
Chapter 4.2-4.9
October 12, 2014
STAT 3375Q Chapter 4.2-4.9
October 12, 2014
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4.2 The Probability Distribution for a Continuous Random Variable
4.2 The Probability Distribution for a Continuous Random Variable
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4.3 Expected Values for
Covariance, Expectation of Linear Combinations, & Conditional Expectation
1 Covariance
1.1 Motivation
While the joint PDF of two random variables fully describes the relationship between two random variables, we would like to have a measurements to summa
Review List
1. The nal Exam will have 6 problems. There are no true or false questions. The only two
proofs are from the list in the end. Another two problems are from the homework.
2. You are allowed to use TWO pages of formula sheet, double sided.
3. Ca
This lecture will focus on Bayes Theorem, but rst we must cover some concepts
1 Preliminaries
First, consider the following situation:
Let A, B1 , B2 , ., Bn be events of the sample space S such that
1. S = B1 B2 . Bn
2. Bi Bj = i, j, where i = j
(i.e. B
1 Calculating Probabilities
To determine the probability of an event we must add up the probabilities of the
individual sample points which are elements of that event
Sometimes guring out the probability of a sample point is not so clear; we need
a syst
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Probability Experiments
Experiments
A probability experiement is an activity that involves chance that
leads to results, that can be repeated
Sample Points
A Sample point is a possible result of a single trial, or repetition, of
an experiment
Event
STAT 3375 Review
1 Taylor Series
1.1 Denition
Let f be a function which is dened and innately dierentiable at all points on some interval (c, d) and let
a be some number in this interval. If f is analytic on this interval, then it can be shown that for al
1
Set Theory Denitions
A set is simply a collection of objects:
The set of counting numbers
The set of students in this classroom
The set of professors in the statistics department
Typical notation: A = cfw_. . .
Examples:
Z = cfw_. . . 2, 1, 0, 1
Outline
6.7 Order Statistics
STAT 3375Q
6.7 Order Statistics
August 18, 2014
STAT 3375Q 6.7 Order Statistics
Outline
6.7 Order Statistics
1 6.7 Order Statistics
STAT 3375Q 6.7 Order Statistics
Outline
6.7 Order Statistics
6.7 Order Statistics
The order st
STAT 3375 Review of Calculus and Some Basic Concepts
1 Dierentiation
1.1 Denitions
Let f be a function which is dened on some interval (c, d) and let a be some number in this interval. The
derivative of the function f at a is the value of the limit
f (a)
Outline 6.3 The Method of Distribution Functions 6.4 The Method of Transformations 6.5 The Method of Moment-Generatin
STAT 3375Q
6.3 6.5
December 2, 2014
STAT 3375Q 6.3 6.5
Outline 6.3 The Method of Distribution Functions 6.4 The Method of Transformations
1 Random Variables
Up until now, we have discussed probability in terms of events and samples spaces
of probability experiments
Anothe way we can discuss probability is in terms of Random Variables
Random Variables
A Random Variable is a numerical rep
Bernoulli & Binomial Distributions
1 Revisiting the Coin Flip
Consider the experiment where we ip a fair coin
Sample space cfw_H, T
Can we make a random variable for this experiment?
Let X = # of heads we observe when we ip a coin
X is 1 if we get a
Indepence & Expectation
1 Indepence
1.1 Motivation
Much like how we discussed the concept of two events being independent of each
other, we can also discuss the idea of random variables being independent
When we say that two random variables are indepen
Marginal & Conditional Distributions
1 Motivation
We began this section of the course discussing the situation in which we take
multiple measurements at the same time and would like to model how the relate
to each other
In these cases sometimes we would
Multivariate Distributions
1 Motivation
Often when we are studying a particular topic we will take various measures that
pertain to that topic
From hre the goal is usually to examine these measures and see what they tell us
about the topic we are studyi
The Gamma Distirbution
1 Denition & Motivation
Our last continuous Distribution is the Beta distribution
The Beta distribution is very useful for modeling bounded random variables with
a non uniform distribution of probability
The Beta distribution is
The Gamma Distirbution
1 Denition & Motivation
Often times we would like to model a continuous Random variable that only has
a positive support
In these cases, the Normal distribution is not an appropriate model because a Normally distibruted Random Var
The Normal Distribution
1 Denition & Motivation
Known by several names including Normal, Gaussian, Bell curve/distribution (here
we will call it the Normal distirbution), the Normal distribution is probably the
most inuential and impactful distribution i
The Gamma Function
1 Denition
Here we will go over the Gamma Function, a function used to verify the Gamma
distribution and the Normal distribution.
Denition 1. The Gamma Function is the function (dened for non-negative
values) such that
(t) =
xt1 ex dx
Continuous Uniform Random Variables
1 Motivation & Denition
We will begin to examine various Continuous Random variables, and we will start
with one of the Simplest, the continuous Uniform Random Variable
Suppose that you are waiting for the bus and the
Continuous Random Variables
1 Denition
Now that we have discussed Discrete Random Variables we will now move on to
continuous Random Variables
Like Discrete Random Variables, continuous Random Variables are numerical representations of the outcomes of a
Moment Generating functions (MGFs)
1 Denitions
Before we can discuss what a moment generating function is, we must rst dene
what we mean by a moment
Denition 1. Let k be a non-negative integer, and let X be a random variable
with support S and PDF pX (x)
Hypergeometric & Poisson
1 Hypergeometric
1.1 Description & Denition
Earlier we discussed the binomial distribution in terms of ipping a coin repeatedly,
or ipping multiple coins at the same time. Either way it was done such that the
ips were independent
Geometric & Negative Binomial
1 Geometric Distribution
Imagine we are again working with a coin that has a probability of landing heads
up of p where 0 < p < 1
Suppose that instead of ipping the coin a certain number of times like we did to
generate the
Outline
5.8 The Expected Value and Variance of Linear Functions of Random Variables
STAT 3375Q
5.8 and 5.11
August 18, 2014
STAT 3375Q 5.8 and 5.11
5.11 Conditional Expectations
Outline
5.8 The Expected Value and Variance of Linear Functions of Random Var
Outline
5.3 Marginal and Conditional Probability Distributions
STAT 3375Q
5.3-5.4
August 18, 2014
STAT 3375Q 5.3-5.4
5.4 Independent Random Variables
Outline
5.3 Marginal and Conditional Probability Distributions
5.4 Independent Random Variables
1 5.3 Mar
Outline
5.2 Bivariate and Multivariate Probability Distributions
STAT 3375Q
5.2
STAT 3375Q 5.2
Outline
5.2 Bivariate and Multivariate Probability Distributions
1 5.2 Bivariate and Multivariate Probability Distributions
5.2 Bivariate and Multivariate Proba