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 the experiment is
completed.
Some experiments have a mo
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 given F is dened by:
P(E | F ) =
Kenneth Harris
kaharri@
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 the events are not mutually inclusive.
The method is c
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 cannot be predicted ahead of time.
The probability mode
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
outcome which cannot be predicted ahead of time.
The p
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
At least one six on 4 throws of a die.
2
Kenneth Harris
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 success or failure. Suppose
the probability of success is p
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
Math 425
Introduction to Probability
Lecture 36
2
an a
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 probability density function of X given
that Y = y was
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
P cfw_X = 0 = P cfw_X = 1 = P cfw_X = 1 =
1
3
and dene
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
Introduction to Probability
Lecture 34
pX |Y (x |y )
=
P(X =
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 , d
constants, then
E [cX + dY ] = cE [X ] + dE [Y ].
Ken
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 (x , y ). Let Z be a random variable
determined by X an
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 are sat in a circle?
Math 425
Introduction to Probabilit
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 some region A of the real line.
Suppose g maps A onto B ,
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 Y = y by
Math 425
Intro to Probability
Lecture 29
pX |Y
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 ).
The sum X + Y is a continuous random variable with dens
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 only nonnegative integers.
What is the probability mas
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
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Equivalently,
P(E | F ) = P cfw_E .
Department of
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 time.
A study records the age, weight, blood pressure, a
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
Kenneth Harris
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has two distinct real r
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 another. The average
number of events in a unit of time is
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 normally distributed, with parameters (the
average score)
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 derived from even a single distribution, such as U , a
u
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 is often easier to nd then the density function.
Kenneth
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 following procedure:
10!
(105)!
1
Choose 5 players out of 10
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 numbers Ap = cfw_a1 , a2 , a3 , . . .
satisfying the foll
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 Prussian
army corps. He collected data from 14 corps ov
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
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=
lim
n
e
x
=
Department of Mathematics
University of Michigan
1+
k
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 following
Kenneth Harris
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1
Only two poss