Lecture 5: Joint Probability Distributions

Bo LI

School of Economics and Management

Tsinghua University

libo@sem.tsinghua.edu.cn

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 Overview

1 Jointly Distributed Random Variables

2 Sampling Distributions and Estimation

The Law of Large Numbers
The Mean, Variance and Moment Generating Functions for
Several Variables
Distributions Based on a Normal Random Sample

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 Jointly Distributed Random Variables

Joint Probability Mass Function

Let X and Y be two discrete rv’s defined on the sample space S

of an experiment. Th joint probability mass function p(x, y ) is defined
for each pair of numbers (x, y ) by

p(x, y) = P(X = x and Y = y)

Let A be any set consisting of pairs of (x, y ) values. Then the

probability that the random pair (X , Y ) lies in A is obtained by
summing the joint pmf over pairs in A:
X X
P[(X , Y ) ∈ A] = p(x, y )
(x,y)∈A

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 Jointly Distributed Random Variables

Joint Probability Table

An insurance agency services customers who have purchsed both
a homeowner’s policy and an automobile policy. For each type of
policy, a deductible amount must be specified.
For an automobile policy, the choices are $100 and $250,
whereas for a homeowner’s policy, the choices are 0,$100 and $200.
Suppose a customer is selected at random from the agency’s files. Let
X be the deductible amount on the auto policy and Y on the
homeowner’s policy.
The joint pmf specifies the probability associated with possible
(X , Y ) pairs, with any other pair having probability zero. Suppose the
joint pmf is given in the accompanying joint probability table:
y
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 Jointly Distributed Random Variables

Marginal Probability Mass Functions

The marginal probability mass functions of X and of Y ,

denoted poy pX (x) and pY (y), respectively, are given by
X X
pX (x) = p(x, y) pY (y ) = p(x, y)
y x

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 Jointly Distributed Random Variables

Calculate Marginal Probability

The possible X values are x = 100 and x = 250, so computing
row totals in the joint probability table yields

pX (100) = p(100, 0) + p(100, 100) + p(100, 200) = .50

pX (250) = p(250, 0) + p(250, 100) + p(250, 200) = .50

The marginal pmf of X is then
(
.5 x = 100, 250
pX (x) =
0 otherwise
Similarly, the marginal pmf of Y is obtained from column totals as

.25 y = 0, 100


pX (x) = .50 y = 200


 0 otherwise
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 Jointly Distributed Random Variables

Joint Probability Density Function

Let X and Y be continuous rv’s. Then f(x,y) is the joint

probability density function for X and Y if for any two-dimensional
set A
Z Z
P[(X , Y ) ∈ A] = f (x, y)dx dy
A
In particular, if A is the two-dimensional rectangle
{(x, y ) : a ≤ x ≤ b, c ≤ y ≤ d}, then

Z b Z d
P[(X , Y ) ∈ A] = P(a ≤ X ≤ b, c ≤ Y ≤ d) = f (x, y )dx dy
a c

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 Jointly Distributed Random Variables

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 Jointly Distributed Random Variables

Marginal Probability Density Functions

The marginal probability density function of X and Y , denoted

by fX (x) and fY (y), respectively, are given by
Z ∞
fX (x) = f (x, y )dy for − ∞ < x < ∞
−∞
Z ∞
fY (y) = f (x, y)dx for − ∞ < y < ∞
−∞

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 Jointly Distributed Random Variables

Marginal Probability Density Functions: Example2

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 Jointly Distributed Random Variables

Marginal Probability Density Functions: Example2

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 Jointly Distributed Random Variables

Marginal Probability Density Functions: Example2

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 Jointly Distributed Random Variables

Independence

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 Jointly Distributed Random Variables

Independence

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 Jointly Distributed Random Variables

Independence

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 Jointly Distributed Random Variables

More than Two Random Variables

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 Jointly Distributed Random Variables

Multinomial Distribution

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 Jointly Distributed Random Variables

Expectation of More than Two Random Variables

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 Jointly Distributed Random Variables

Expectation of More than Two Random Variables:

Example

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 Jointly Distributed Random Variables

Covariance

The covariance between two rv’s X and Y is

Cov(X , Y ) = E [(X − µX ) (Y − µY )]
 P P
y (x − µX ) (y − µY ) p(x, y ),



 x

 if X and Y are discrete
= R∞ R∞
−∞ −∞ (x − µX ) (y − µY ) f (x, y )dx dy ,




if X and Y are continuous



Cov(X , Y ) = E(XY ) − µX · µY

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 Jointly Distributed Random Variables

Expected Values, Cov. Corr.: Example

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 Jointly Distributed Random Variables

Expected Values, Cov. Corr.: Example

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 Jointly Distributed Random Variables

Application: Portfolios and Random Variables

How should money be allocated among several stocks that form a

portfolio?
Need to manipulate several random variables at once to
understand portfolios
Since stocks tend to rise and fall together, random variables for
these events must capture dependence

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 Jointly Distributed Random Variables

Application: Portfolios and Random Variables

Two Random Variables

Suppose a day trader can buy stock in two companies, IBM and
Microsoft, at $100 per share
X denotes the change in value of IBM
Y denotes the change in value of Microsoft

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 Jointly Distributed Random Variables

Application: Portfolios and Random Variables

Comparisons and the Sharpe Ratio

The day trader can invest $200 in

Two shares of IBM;
Two shares of Microsoft; or
One share of each

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 Jointly Distributed Random Variables

Application: Portfolios and Random Variables

Joint Probability Distribution of X and Y

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 Jointly Distributed Random Variables

Application: Portfolios and Random Variables

Probability Distribution for the Two Stocks

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 Jointly Distributed Random Variables

Application: Portfolios and Random Variables

Dependent Random Variables

Joint probability table shows changes in values of IBM and
Microsoft (X and Y ) are dependent
The dependence between them is positive

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 Jointly Distributed Random Variables

Application: Portfolios and Random Variables

Which portfolio should she choose?

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 Jointly Distributed Random Variables

Application: Portfolios and Random Variables

Sharpe Ratio for Mixed Portfolio

(µ + µY ) − 2rf 0.22 − 0.03

S(X + Y ) = pX ≈ √ ≈ 0.050
Var(X + Y ) 14.64

Summary of Sharpe Ratios (Advantage of Diversifying)

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 Jointly Distributed Random Variables

Linear Combination of Random Variables

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 Jointly Distributed Random Variables

Correlation

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 Jointly Distributed Random Variables

Correlation

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 Jointly Distributed Random Variables

Association and Causation

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 Jointly Distributed Random Variables

Conditional Distributions

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 Jointly Distributed Random Variables

Conditional Distributions: Example1

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 Jointly Distributed Random Variables

Conditional Distributions: Example1

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 Jointly Distributed Random Variables

Conditional Distributions: Example2

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 Jointly Distributed Random Variables

Conditional Distributions: Example2

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 Jointly Distributed Random Variables

Conditional Mean/Conditional Expectation

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 Jointly Distributed Random Variables

Conditional Variance
The conditional mean of any function g(Y ) can be obtained similarly. In
the discrete case,
X
E(g(Y )|X = x) = g(y )pY |X (y |x)
y ∈DY

In the continuous case

Z ∞
E(g(Y )|X = x) = g(y )fY |X (y |x)dy
−∞

The conditional variance of Y given X = x is

σY2 |X =x = V (Y |X = x) = E [Y − E(Y |X = x)]2 |X = x



There is a shortcut formula for the conditional variance analogous to that for
V (Y ) itself:

σY2 |X =x = V (Y |X = x) = E Y 2 |X = x − µ2Y |X =x

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 Jointly Distributed Random Variables

Conditional Distributions

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 Jointly Distributed Random Variables

Conditional Distributions

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 Jointly Distributed Random Variables

Conditional Distributions

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 Jointly Distributed Random Variables

Conditional Distributions

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 Jointly Distributed Random Variables

Conditional Distributions

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 Jointly Distributed Random Variables

The Bivariate Normal Distribution

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 Jointly Distributed Random Variables

The Bivariate Normal Distribution

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 Jointly Distributed Random Variables

Joint Normal Distribution (Density)

 
−1/2 1 0 −1
fY (y) = det(2π · Σ) exp − (y − µ) Σ (y − µ)
2

Bivariate normal densities with µX = µY = 0 and σX = σY = 1

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 Jointly Distributed Random Variables

The Bivariate Normal Distribution

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 Jointly Distributed Random Variables

The Bivariate Normal Distribution

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 Jointly Distributed Random Variables

Regression to the Mean

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 Jointly Distributed Random Variables

Regression to the Mean

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 Jointly Distributed Random Variables

The Bivariate Normal Distribution

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 Jointly Distributed Random Variables

Conditional Mean and Variance as Random

Variables: A Key Theorem

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 Jointly Distributed Random Variables

Conditional Expectations: Example1

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 Jointly Distributed Random Variables

Conditional Expectations: Example2

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 Jointly Distributed Random Variables

Conditional Expectations: Random Sums

PN
This example introduces sums of the type T = i=1 Xi , where N is a
random variable with a finite expectation and Xi are random variables that are
independent of N and have the common mean E(X ).
An insurance company might receive N claims in a given period of time,
and the amounts of the individual claims might be modeled as random
variables X1 , X2 , · · · The random variable N could denote the number of
customers entering a store and Xi the expenditure of the i th customer, or N
could denote the number of jobs in a single-server queue and Xi the service
time for the ith job. For this last case, T is the time to serve all the jobs in the
queue.
According to Theorem a, E(T ) = E[E(T |N)]. Since
E(T |N = n) = nE(X ), i.e., E(T |N) = NE(X ) and thus

E(T ) = E[NE(x)] = E(N)E(X )

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 Jointly Distributed Random Variables

Conditional Expectations: Random Sums

Assume Xi are independent random variables with the same
mean, and the same variable, Var(X ), and that Var(N) < ∞ .
According to Theorem b ,
Var(T ) = E[Var(T |N)] + Var[E(T |N)]

Because E(T |N) = NE(X ), we have

Var[E(T |N)] = [E(X )]2 Var(N)
Pn

Also, since Var(T |N = n) = Var i=1 Xi = n Var(X ), we have
Var(T |N) = N Var(X ). Further
E[Var(T |N)] = E(N) Var(X )

We thus obtain
Var(T ) = [E(X )]2 Var(N) + E(N) Var(X )
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 Jointly Distributed Random Variables

Conditional Expectations: Random Sums

As a concrete example, suppose that the number of insurance

claims in a certain time period has expected value equal to 900 and
standard deviation equal to 30, as would be the case if the number
were Poisson random variable with expected value 900. Suppose that
the average claim value is $1000 and the standard deviation if $500.
Then the expected value of the total, T , of the claims is
E(T ) = $900, 000 and the variance of T is

Var(T ) = 10002 × 900 + 900 × 5002 = 1.125 × 109

Or the standard deviation of T is $33, 541

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 Jointly Distributed Random Variables

Conditional Expectations: Random Sums

The insurance company could then plan on total claims of

$900, 000 plus or minus a few standard deviations (by Chebyshev’s
inequality). Observe that if the total number of claims were not variable
but were fixed at N = 900 , the variance of the total claims would be
given by E(N) Var(X ) in the previous expression. The result would be
a standard deviation equal to $15, 000. The variability in the number of
claims thus contributes substantially to the uncertainty in the total.

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 Jointly Distributed Random Variables

Prediction
Suppose we want the predict Y using an instrument X . Our
predictor is hence denoted as h(X ). We need some measure of the
effectiveness of a prediction. One that is amenable to mathematical
analysis and that is widely used is the mean square error (MSE):
h i
MSE = E [Y − h(X ))2

Note that
h i
MSE = E[[Y − E(Y |X ))]2 + E (E(Y |X ) − h(X ))2

Thus the minimization function h∗ (X ) is

h∗ (X ) = E(Y |X )

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 Jointly Distributed Random Variables

Prediction

For the bivariate normal distribution, we found that

σY
E(Y |X ) = µY + ρ (X − µX )
σX
This linear function of X is thus the minimum mean squared error
predictor of Y from X . It can be shown that for general joint
distribution of Y and X , the best linear predictor of Y in terms of X
(having the form α + βX ) in the sense of minimum MSE is also
σY
E(Y |X ) = µY + ρ (X − µX )
σX
Note that the optimal linear predictor depends on the joint distribution
of X and Y only through their means, variances, and covariance.
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 Sampling Distributions and Estimation The Law of Large Numbers

The Law of Large Numbers(LLN)

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 Sampling Distributions and Estimation The Law of Large Numbers

Two Inequalities about Expectation and Variance

Markov Inequality: Assume X is a nonnegative r.V. and for which

E(X ) exists, then
P(X >= t) <= E(X )/t

Chebyshev Inequality

P(|X − E(X )| >= t) <= Var(X )|t 2 2

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 Sampling Distributions and Estimation The Law of Large Numbers

Proof of the Law of Large Numbers

Proof using Chebyshevs inequality

Var(X ) σ2
P(|X − µ| > ) ≤ = →0
2 n2

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 The Mean, Variance and Moment Generating Functions for
Sampling Distributions and Estimation Several Variables

Linear Combination of Several Variables: Mean

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 The Mean, Variance and Moment Generating Functions for
Sampling Distributions and Estimation Several Variables

Linear Combination of Several Variables:

Variance

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 The Mean, Variance and Moment Generating Functions for
Sampling Distributions and Estimation Several Variables

Linear Combination of Several Variables:

Example

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 The Mean, Variance and Moment Generating Functions for
Sampling Distributions and Estimation Several Variables

Linear Combination of Several Variables: Normal

Variables

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 The Mean, Variance and Moment Generating Functions for
Sampling Distributions and Estimation Several Variables

A Proof Using MGF

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 The Mean, Variance and Moment Generating Functions for
Sampling Distributions and Estimation Several Variables

Linear Combination of Several Variables:

Example cont’d

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 Sampling Distributions and Estimation Distributions Based on a Normal Random Sample

Chisquare Distribution

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 Sampling Distributions and Estimation Distributions Based on a Normal Random Sample

Chisquare Distribution

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 Sampling Distributions and Estimation Distributions Based on a Normal Random Sample

Chisquare Distribution

If X1 ∼ χ2v1 , X2 ∼ χ2v2 , and they are independent, then

X1 + X2 ∼ χ2v1 +v2
If Z1 , Z2 , . . . , Zn are independent and each has the standard
normal distribution, then Z12 + Z22 + · · · + Zn2 ∼ χ2n
If X1 , X2 , . . . , Xn are a random sample from a normal distribution,
then X and S 2 are independent.
If X1 , X2 , . . . , Xn are a random sample from a normal distribution,
then (n − 1)S 2 /σ 2 ∼ χ2n−1 .

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 Sampling Distributions and Estimation Distributions Based on a Normal Random Sample

t distribution

Let Z be a standard normal rv and let X be a χ2v rv independent of

Z . Then the t distribution with degrees of freedom v is defined to be
the distribution of the ratio
Z
T =p
X /v

Sometimes we will include a subscript to indicate the df, t = tv

If X1 , X2 , . . . , Xn is a random sample from a normal distribution
N µ, σ 2 , then

X −µ
T = √
S/ n
has the t distribution with (n − 1) degrees of freedom, tn−1
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 Sampling Distributions and Estimation Distributions Based on a Normal Random Sample

t distribution

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 Sampling Distributions and Estimation Distributions Based on a Normal Random Sample

F distribution

Let X1 and X2 be independent chi-squared random variables with

v1 and v2 degrees of freedom, respectively. The F distribution v1
numerator degrees of freedom and v2 denominator degrees of
freedom is defined to be the distribution of the ratio
X1 /v1
F =
X2 /v2

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 Sampling Distributions and Estimation Distributions Based on a Normal Random Sample

F distribution

Suppose that we have a random sample of m observations from

the normal population N µ1 , σ12 and an independent random sample

of n observations from al second normal population N µ2 , σ22 . Then

for the sample variance from the first group we know (m − 1)S12 /σ12 is
χ2m−1 , and similarly for the second group (n − 1)S22 /σ22 is χ2n−1 . Thus,

(m−1)S12 /σ12
m−1 S12 /σ12
Fm−1,n−1 = =
(n−1)S12 /σ22 S22 /σ22
n−1

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 Sampling Distributions and Estimation Distributions Based on a Normal Random Sample

F distribution

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