Jointly Distributed Random Variables
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)
and the probability for each outcome is
1
36 .
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,
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 ),
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
1
of that random pair (x, y ) is 36
.
Jointly Distributed Random Variables
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
Jointly Distributed Random Variables
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
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?
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?
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?
Jointly Distributed Random Variables
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
x
Jointly Distributed Random Variables
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
Jointly Distributed Random Variables
Jointly Distributed Random Variables
Definition
Let X and Y be continuous random variables. A joint probability
density function f (x, y ) for
R ∞these
R ∞ two variables is a function
satisfying 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) =
d
f (x, y )dydx
a
c
Jointly Distributed Random Variables
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 < ∞
−∞
Jointly Distributed Random Variables
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
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?
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 ?
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?
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?