Transcript Lecture 4

Today
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Today: More of Chapter 2
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Reading:
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Assignment #2 is up on the web site
www.stat.lsa.umich.edu/~dbingham/stat405
Please read Chapter 2
Suggested problems: 2.4, 2.5, 2.7, 2.13, 2.25, 2.28, 2.32, 2R1, 2R2
Random Variables
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Often interested in a characteristic that varies from one individual to
another
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A characteristic of the outcome of a random experiment is called a
random variable (RV)
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RV’s are numeric or categorical
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Have discrete and continuous RV’s
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Usually use capital letters to denote an RV and small letters to denote
outcomes
Random Variables
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More formally:
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For a given model, with sample space, Ω , a random variable, X(w), is a
function from the sample space to the real numbers (where w is in the
sample space)
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Each possible value, x, of a random variable, X, is an event
(collection of outcomes from the sample space)
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P( X ( w)  x)  P(set of w' s with X ( w)  x) 
 P(w)
X ( w)  x
Random Variables
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Example:
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If two balanced die are rolled, then an outcome is a pair w=(i,j), where i
and j are integers between 1 and 6
If X(w)=i+j, then this represents the random variable that computes the
sum of the dice
P(X(w)=7) =
Discrete Random Variable
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A random variable, X, is said to be discrete if its possible values may
be arranged in a sequence, {x1, x2,…}
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E.g., X is the outcome of a roll of a die
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E.g., If X has non-negative integer values
Probability Function
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The probability function of a discrete random variable X(w) is
f ( x)  P( X ( w)  x) 
 P(w)
X ( w)  x
Example
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If two balanced die are rolled, then an outcome is a pair w=(i,j),
where i and j are integers between 1 and 6
If X(w)=i+j, then this represents the random variable that computes
the sum of the dice
f(x)=
Example (2.1c)
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A coin is tossed until a heads appears
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The possible outcomes are Ω=
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Let X be the random variable denoting the trial for which the first
heads is observed
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f(x)=
Properties of Probability Mass Functions
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f ( x)  0 for all x

f ( x)  1
x in 
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Probability function demonstrates how the probability is distributed
among the possible values of the random variable
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We refer to how it is distributed as the probability distribution
Joint Distributions
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Understanding the relationship among variables defined on the same
sample space can be quite important
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If two random variables, X and Y, are defined on the same sample
space, they are said to be jointly distributed
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To study their relationship we consider them together as a random
vector (X,Y)
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The values of the random vector (x,y) have a joint probability (mass)
function f(x,y)=P(X(w)=x,Y(w)=y)
Properties of the Joint Probability Mass Functions
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f ( x, y)  0 for all x, for all y
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
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f ( x, y )  1
( x, y ) in 
f ( x) 

y in 
f ( x, y )  1
Example (2.6)
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From a bowl containing 5 poker chips labeled 1-5, two are selected
at random, one at a time, with replacement
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Let X denote the RV that describes the number on the first draw and
Y denote the denote the RV that describes the number on the second
draw
Construct a joint probability table to display the distribution
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Example (2.6)
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Find P(X=4,Y=5)
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Find P(X+Y=5)
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Find P(|X-Y|=2)
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Find P(X=1, Y=2 or 4)
Example (2.8)
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In a particular population, there may be a relationship between
education and opinion on the death penalty
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The joint probabilities for the random variables education (X) and
Opinion (Y) are given in the table below
Opinion
Education
Grade School
High School
College
Favor
0.15
0.20
0.25
Oppose
0.05
0.10
0.25
Example (2.8)
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What is the probability that a randomly selected person has a high
school education and opposes gun control?
Marginal Distributions
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Given the joint distribution of two random variables, the distribution
of one of the variables alone is called the marginal distribution
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Marginal Distributions of X and Y:
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f X ( x)  P ( X  x) 
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fY ( y )  P(Y  y ) 

f ( x, y )
y in 

x in 
f ( x, y )
Example (2.6)
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Find P(Y=2)
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Find the marginal distribution of X
Example (2.8)
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Give the probability distribution for the variable Education
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Give the probability distribution for the variable Opinion
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Given that you know that the randomly selected individual has a
college degree. What is the probability that they favor the death
penalty
Independence
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Random variables X and Y are independent iff for every pair (x,y),
f(x,y)=f(x)f(y)
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Same notion as before
Example (2.6)
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Are the variables X and Y independent
Example (2.8)
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Are the random variables Education and Opinion independent