Jointly Distributed Random Variables
Consider tossing a fair
(1,1) (1,2) (1,3)
(2,1) (2,2) (2,3)
···
···
···
(6,1) (6,2) (6,3)
die twice. Then the outcomes would be
(1,4) (1,5) (1,6)
(2,4) (2,5) (2,6)
···
···
···
(6,4) (6,5) (6,6)
1
.
and the probability for each outcome is 36
If we define two random variables by X = the outcome of the first
toss and Y = the outcome of the second toss, then the outcome
for this experiment (two tosses) can be describe by the random pair
(X , Y ), and the probability for any possible value of that random pair
1
(x, y ) is 36
.
Liang Zhang (UofU)
Applied Statistics I
July 3, 2008
1 / 21
Jointly Distributed Random Variables
Definition
Let X and Y be two discrete random variables defined on the sample
space S of an experiment. The 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 )
P P
(It must be the case that p(x, y ) ≥ 0 and x y p(x, y ) = 1.)
For any event A consisting of pairs of (x, y ), the probability P[(X , Y ) ∈ A]
is obtained by summing the joint pmf over pairs in A:
XX
P[(X , Y ) ∈ A] =
p(x, y )
(x,y )∈A
Liang Zhang (UofU)
Applied Statistics I
July 3, 2008
2 / 21
Jointly Distributed Random Variables
Example (Problem 75)
A restaurant serves three fixed-price dinners costing $12, $15, and $20.
For a randomly selected couple dinning at this restaurant, let X = the
cost of the man’s dinner and Y = the cost of the woman’s
dinner. If the joint pmf of X and Y is assumed to be
y
p(x, y )
12 15 20
12 .05 .05 .10
x
15 .05 .10 .35
20 0 .20 .10
a. What is the probability for them to both have the $12 dinner?
b. What is the probability that they have the same price dinner?
c. What is the probability that the man’s dinner cost $12?
Liang Zhang (UofU)
Applied Statistics I
July 3, 2008
3 / 21
Jointly Distributed Random Variables
Definition
Let X and Y be two discrete random variables defined on the sample
space S of an experiment with joint probability mass function p(x, y ).
Then the pmf’s of each one of the variables alone are called the marginal
probability mass functions, denoted by pX (x) and pY (y ), respectively.
Furthermore,
X
X
pX (x) =
p(x, y ) and pY (y ) =
p(x, y )
y
Liang Zhang (UofU)
x
Applied Statistics I
July 3, 2008
4 / 21
Jointly Distributed Random Variables
Example (Problem 75) continued:
The marginal probability mass functions for the previous example is
calculated as following:
y
p(x, y ) 12 15 20 p(x, ·)
12
.05 .05 .10
.20
x
15
.05 .10 .35
.50
20
0 .20 .10
.30
p(·, y ) .10 .35 .55
Liang Zhang (UofU)
Applied Statistics I
July 3, 2008
5 / 21
Jointly Distributed Random Variables
Definition
Let X and Y be continuous random variables. A joint probability
density functionR f (x,
y ) for these two variables is a function satisfying
∞ R∞
f (x, y ) ≥ 0 and −∞ −∞ f (x, y )dxdy = 1.
For any two-dimensional set A
ZZ
P[(X , Y ) ∈ A] =
f (x, y )dxdy
A
In particular, if A is the two-dimensilnal rectangle
{(x, y ) : a ≤ x ≤ b, c ≤ y ≤ d}, then
Z bZ
P[(X , Y ) ∈ A] = P(a ≤ X ≤ b, c ≤ Y ≤ d) =
f (x, y )dydx
a
Liang Zhang (UofU)
Applied Statistics I
d
c
July 3, 2008
6 / 21
Jointly Distributed Random Variables
Definition
Let X and Y be continuous random variables with joint pdf f (x, y ). Then
the marginal probability density functions 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 < ∞
−∞
Liang Zhang (UofU)
Applied Statistics I
July 3, 2008
7 / 21
Jointly Distributed Random Variables
Example (variant of Problem 12)
Two components of a minicomputer have the following joint pdf for their
useful lifetimes X and Y :
(
xe −(x+y ) x ≥ 0 and y ≥ 0
f (x, y ) =
0
otherwise
a. What is the probability that the lifetimes of both components excceed
3?
b. What are the marginal pdf’s of X and Y ?
c. What is the probability that the lifetime X of the first component
excceeds 3?
d. What is the probability that the lifetime of at least one component
excceeds 3?
Liang Zhang (UofU)
Applied Statistics I
July 3, 2008
8 / 21
Jointly Distributed Random Variables
Example (Problem 17)
An ecologist wishes to select a point inside a circular sampling region
according to a uniform distribution (in practice this could be done by first
selecting a direction and then a distance from the center in that direction).
let X = the x coordinate of the point selected and Y = the y
coordinate of the point selected. If the circle is centered at (0,0)
and has radius R, then the joint pdf of X and Y is
(
1
x2 + y 2 ≤ R2
2
f (x, y ) = πR
0
otherwise
What is the probability that the x coordinate of the selected point is
within R/2 of the center of the circular region?
Liang Zhang (UofU)
Applied Statistics I
July 3, 2008
9 / 21
Jointly Distributed Random Variables
Recall the following example (variant of Problem 12):
Two components of a minicomputer have the following joint pdf for their
useful lifetimes X and Y :
(
xe −(x+y ) x ≥ 0 and y ≥ 0
f (x, y ) =
0
otherwise
The marginal pdf’s of X and Y are
fX (x) = xe −x and fY (y ) = e −y ,
respectively. We see that
fX (x) · fY (y ) = f (x, y ).
Liang Zhang (UofU)
Applied Statistics I
July 3, 2008
10 / 21
Jointly Distributed Random Variables
Definition
Two random variables X and Y are said to be independent if for every
pair of x and y values,
p(x, y ) = pX (x) · pY (y )
when X and Y are discrete
or
f (x, y ) = fX (x) · fY (y )
when X and Y are continuous
If the above relation is not satisfied for all (x, y ), then X and Y are said to
be dependent.
Liang Zhang (UofU)
Applied Statistics I
July 3, 2008
11 / 21
Jointly Distributed Random Variables
Examples:
Problem 12. The joint pdf for X and Y is
(
xe −x(1+y ) x ≥ 0 and y ≥ 0
f (x, y ) =
0
otherwise
The marginal pdf’s of X and Y are
fX (x) = −e −x and fY (y ) =
1
,
(1 + y )2
respectively. We see that
fX (x) · fY (y ) 6= f (x, y ).
Liang Zhang (UofU)
Applied Statistics I
July 3, 2008
12 / 21
Jointly Distributed Random Variables
Examples:
Our first example: tossing a fair die. If we let X = the outcome of the
first toss and Y = the outcome of the second toss, then we will
have
p(x, y ) = pX (x) · pY (y )
Obviously, we know the two toss should be independent.
Our second example: dinning choices.
y
p(x, y ) 12 15 20 p(x, ·)
12
.05 .05 .10
.20
pX (12) · pY (12) 6= p(12, 12).
x
15
.05 .10 .35
.50
20
0 .20 .10
.30
p(·, y ) .10 .35 .55
Liang Zhang (UofU)
Applied Statistics I
July 3, 2008
13 / 21
Jointly Distributed Random Variables
Definition
If X1 , X2 , . . . , Xn are all discrete random variables, the joint pmf of the
variables is the function
p(x1 , x2 , . . . , xn ) = P(X1 = x1 , X2 = x2 , . . . , Xn = xn )
If the random variables are continuous, the joint pdf of X1 , X2 , . . . , Xn is
the function f (x1 , x2 , . . . , xn ) such that for any n intervals
[a1 , b1 ], . . . , [an , bn ],
P(a1 ≤ X1 ≤ b1 , . . . an ≤ Xn ≤ bn ) =
Z b1 Z bn
···
f (x1 , . . . , xn )dxn . . . dx1
a1
Liang Zhang (UofU)
an
Applied Statistics I
July 3, 2008
14 / 21
Jointly Distributed Random Variables
Example:
Consider tossing a particular die six times. The probabilities for outcomes
of each toss are given as following:
1
2
3
4
5
6
x
p(x) .15 .20 .25 .20 .15 .05
If we are interested in obtaining exactly three “1’s”, then this experiment
can be modeled by the binomial distribution.
However, if the question is “what is the probability for obtaining exactly
three 1’s, two 5’s and one 6”, the binomial distribution can not do
the job.
Let Xi = number of i’s from the experiment (six tosses). Then
P(X1 = 3, X2 = 0, X3 = 0, X4 = 0, X5 = 2, X6 = 1) =
6!
(.15)3 (.15)2 (.05)1
3!2!1!
Liang Zhang (UofU)
Applied Statistics I
July 3, 2008
15 / 21
Jointly Distributed Random Variables
Multinomial Distribution:
1. The experiment consists of a sequence of n trials, where n is fixed in
advance of the experiment;
2. Each trial can result in one of the r possible outcomes;
3. The trials are independent;
3. The trials are identical, which means the probabilities for outcomes of
eachP
trial are the same. We use p1 , p2 , . . . , pr to denote them. (pi > 0
and ri=1 pi = 1)
Definition
An experiment for which Conditions 1 — 4 are satisfied is called a
multinomial experiment.
Let Xi = the number of trials resulting in outcome i, then the
joint pmf of X1 , X2 , . . . , Xr is called a multinomial distribution.
Liang Zhang (UofU)
Applied Statistics I
July 3, 2008
16 / 21
Jointly Distributed Random Variables
Remark:
The joint pmf is
p(x1 , x2 , . . . , xr ) =
(
x1 x2
n!
xr
(x1 !)(x2 !)···(xr !) p1 p2 · · · pr
0
Liang Zhang (UofU)
0 ≤ xi ≤ n with
Pr
i=1 xi
=n
otherwise
Applied Statistics I
July 3, 2008
17 / 21
Jointly Distributed Random Variables
Definition
The random variables X1 , X2 , . . . , Xn are said to be independent if for
every subset Xi1 , Xi2 , . . . , Xik of the variables (each pair, each triple, and so
on), the joint pmf or pdf of the subset is equal to the product of the
marginal pmf’s or pdf’s.
e.g. one way to construct a multinormal distribution is to take the
product of pdf’s of n independent standard normal rv’s:
1 −x22 /2
1 −xn2 /2
1 −x12 /2
√ e
f (x1 , x2 , . . . , xn ) = √ e
··· √ e
2π
2π
2π
1
2
2
2
= √
e −(x1 +x2 +···+xn )/2
( 2π)n
Liang Zhang (UofU)
Applied Statistics I
July 3, 2008
18 / 21
Jointly Distributed Random Variables
Recall the following example (Problem 12):
Two components of a minicomputer have the following joint pdf for their
useful lifetimes X and Y :
(
xe −x(1+y ) x ≥ 0 and y ≥ 0
f (x, y ) =
0
otherwise
If we find out that the lifetime for the second component is 8 (Y = 8),
what is the probability for the first component to have a lifetime more
than 8, i.e. what is P(X ≥ 8 | Y = 8)?
We can answer this question by studying conditional probability
distributions.
Liang Zhang (UofU)
Applied Statistics I
July 3, 2008
19 / 21
Jointly Distributed Random Variables
Definition
Let X and Y be two continuous rv’s with joint pdf f (x, y ) and marginal Y
pdf fY (y ). Then for any Y value y for which fY (y ) > 0, the conditional
probability density function of X given that Y = y is
fX |Y (x | y ) =
f (x, y )
fY (y )
−∞<x <∞
If X and Y are discrete, then conditional probability mass function of
X given that Y = y is
pX |Y (x | y ) =
Liang Zhang (UofU)
p(x, y )
pY (y )
Applied Statistics I
−∞<x <∞
July 3, 2008
20 / 21
Jointly Distributed Random Variables
Example (Problem 12 revisit):
Two components of a minicomputer have the following joint pdf for their
useful lifetimes X and Y :
(
xe −x(1+y ) x ≥ 0 and y ≥ 0
f (x, y ) =
0
otherwise
What is P(X ≥ 8 | Y = 8)?
f (x, y )
fX |Y (x | y ) =
=
fY (y )
(
x(1 + y )2 e −x(1+y )
0
x ≥ 0 and y ≥ 0
otherwise
Then
Z
8
P(X ≥ 8 | Y = 8) = 1 −
−∞
Z 8
fX |Y (x | 8)dx
81xe −9x dx = 73e −72
=1−
0
Liang Zhang (UofU)
Applied Statistics I
July 3, 2008
21 / 21