Conditional Probability
Conditional Probability
Very few experiments amount to just one action with random
Math 425
Introduction to Probability
Lecture 8
outcomes.
Sometimes conditions change before
Review: Conditioning Rule and Conjunctions
Conditional probability
Math 425
Introduction to Probability
Lecture 9
Denition (Conditioning Rule)
Let E and F be events.
The conditional probability of E g
Overview
Unions and Intersections
! This lecture will focus on computing the probability of arbitrary nite
Math 425
Introduction to Probability
Lecture 7
unions of events
P (E1 E2 . . . En )
even when
Review
The probability model
Math 425
Introduction to Probability
Lecture 6
The basis of probability is the experiment:
An experiment is a repeatable procedure that has a a measurable
outcome which c
The probability model
The probability model
Math 425
Introduction to Probability
Lecture 5
The basis of probability is the experiment:
An experiment is a repeatable procedure that has a a measurable
A problem of Probability
A probability problem
Math 425
Introduction to Probability
Lecture 4
Lets begin by gaming and gambling, where the of probability lies.
Problem. Which event is more likely?
1
Example: Coin tossing
Example continued
Consider a Bernoulli trials process with IID indicator variables
Math 425
Intro to Probability
Lecture 37
X1 , X2 , . . . denoting whether the trial was a succ
Bounding Probabilities
Chebyshevs Inequality
A measure of the concentration of a random variable X near its
mean is its variance .
Chebyshevs Inequality says that the probability that X lies outside
Conditional Expectation continuous
Denition Continuous Case
Let X and Y be jointly continuous random variables with joint
Math 425
Intro to Probability
Lecture 35
density f (x , y ). The conditional
Expectation of Products
Expectation of Products
expectations. Consider Xthat an expectation of a product is a product
It is NOT generally true
of
where
Math 425
Introduction to Probability
Lecture 33
Conditional Expectation discrete
Denition Discrete Case
Let X and Y be jointly discreteYrandom variables.cfw_Y =dened the
We
conditional mass of X given that = y (provided P
y > 0) by
Math 425
Intro
Review
Expectation of Sums
The expectation of a sum is a sum of expectations.
Math 425
Introduction to Probability
Lecture 32
Theorem
If X and Y are random variables whose expectations exist and c ,
Expectation of a Function of Random Variables
Joint Expectation
Let X and Y be continuous (discrete) random variables X and Y
Math 425
Intro to Probability
Lecture 31
with joint density (mass) fX ,Y
Review
Problem: Sitting in a circle
Problem. 5 boys and 5 girls are to be assigned seats so that boys and
girls alternate. How many arrangements are there if they are sat in a
line? How many if they a
Introduction One Function of Random Variables
Functions of a Random Variable: Density
Let g(x ) = y be a one-to-one function whose derivative is nonzero
Math 425
Intro to Probability
Lecture 30
on so
Discrete Conditional Probability
Denition: Discrete Conditional Probability
Denition
Let X and Y be discrete random variables on the sample space.
The conditional probability mass function of X given
Convolutions reminder
Theorem: convolutions and sums of r.v.s
Math 425
Intro to Probability
Lecture 28
Theorem
Let X and Y be independent continuous random variables with density
fX (x ) and fY (y ).
Sums of Discrete Random Variables
Problem involving sums
Problem. Let X and Y be two independent discrete random variables
with known probability mass functions pX and pY and whose possible
values are
Independence
Independent Random Variable
Math 425
Intro to Probability
Lecture 26
Recall. Two events E and F are independent if and only if
P cfw_E F = P cfw_E P cfw_F .
Kenneth Harris
[email protected]
Examples of Joint Distributions
Jointly distributed random variables
In many experiments we have several random variables that we are
Math 425
Intro to Probability
Lecture 25
interested at the same t
Example: Roots
Example: Roots
Example
Math 425
Intro to Probability
Lecture 24
Let U be a uniformly distributed random variable on [0, 1]. What is the
probability that the equation
x 2 + 4Ux + 1 = 0
K
Poisson Process
Poisson Process: informal
A Poisson process is an experiment in which events randomly occur
Math 425
Intro to Probability
Lecture 23
continuously in time and independently of one anot
Example: Grading on a curve
Example: Grading on a curve
Example
Math 425
Intro to Probability
Lecture 22
Final exams at Podunk U. are constructed so that the distribution of
scores is approximately no
Functions of a Random Variable
Functions of a Random Variable: Expectation
There are an overwhelming number of possible random variables
Math 425
Introduction to Probability
Lecture 21
which can be d
Cumulative Distribution Functions
Cumulative Distribution Function
There is another way to give the probability of a continuous random
Math 425
Introduction to Probability
Lecture 20
variable, that i
Review
Review
Math 425
Introduction to Probability
Lecture 2
Problem. A coach has ten players and wants to make two teams of
ve. How many ways are there of matching-up 5-on-5?
Method 1. Use the follow
Probability mass functions
Discrete Random Variable
A real-valued function p : R R is a probability mass function if
Math 425
Introduction to Probability
Lecture 19
there is an enumeration of real nu
Rare events
Example
Example
Math 425
Introduction to Probability
Lecture 18
In one of the earliest studies of the Poisson distribution, von
Bortkiewicz (1898) considered deaths from mule kicks in the
Poisson Random Variables
Two equations for ex
Math 425
Introduction to Probability
Lecture 17
We will need the following two ways of specifying e .
x
ex
Kenneth Harris
[email protected]
=
lim
n
e
x
=
Bernoulli trials
Bernoulli Trials
Math 425
Introduction to Probability
Lecture 16
Denition
By a Bernoulli trials process, we mean a sequence of trials (repetitions
of an experiment) satisfying the fol