Chapter 5
Joint Probability Distributions and
Random Samples
Many problems in probability and statistics involve several ran-
dom variables simultaneously. In this chapter, we first discuss
probability models for the joint (i.e., simultaneous) behavior of
several random variables.
Then later in this chapter, we consider functions of
n
random
variables
X
1
, X
2
,
· · ·
, X
n
, focusing especially on their average
1
n
P
n
i
=1
X
i
. We call any such function, itself a random variable,
a
(sample) statistic
.
Methods from probability are used to
obtain information about the distribution of a statistic.
The premier result of this type is the
Central Limit Theorem
(CLT)
, the basis for many inferential procedures involving large
sample sizes.
5.1 Jointly Distributed Random Variables
Here we start with joint probability distributions for two (dis-
crete or continuous) random variables.
94

Chapter 5. Joint Probability Distributions and Random Samples
STAT 155
Two Discrete Random Variables
Suppose
X
and
Y
are both discrete random variables. The
joint probability mass function
p
(
x, y
)
is defined for each pair
(
x, y
)
by
p
(
x, y
) =
P
(
X
=
x, Y
=
y
)
satisfying
p
(
x, y
)
≥
0
and
P
x
P
y
p
(
x, y
) = 1
.
Now let
A
be any set consisting of pairs of
(
x, y
)
values (e.g.,
A
=
{
(
x, y
) :
x
+
y
= 5
}
or
{
(
x, y
) : max(
x, y
)
3
}
).
Then the probability
P
[(
X, Y
)
2
A
]
is obtained by summing the
joint pmf over pairs in
A
:
P
[(
X, Y
)
2
A
] =
X X
(
x,y
)
2
A
p
(
x, y
)
The
marginal probability mass function
of
X
and
Y
, denoted
by
p
X
(
x
)
and
p
Y
(
y
)
, respectively, are given by
p
X
(
x
) =
P
(
X
=
x
) =
X
y
p
(
x, y
)
for each possible value
x
p
Y
(
y
) =
P
(
Y
=
y
) =
X
x
p
(
x, y
)
for each possible value
y
95

Chapter 5. Joint Probability Distributions and Random Samples
STAT 155
ExampleSuppose a fair coin is tossed three times. The sample space isS={HHH, HHT, HTT, HTH, TTT, TTH, THH, THT}.Let X denote the total number of heads; and Y denote the number of headon the first toss.(a) Write out the joint probability mass functionp(x, y)for each possiblepair ofxandyvalues.(b) CalculateP(X-Y= 1).(c) What is the marginal probabilitypX(x)andpY(y)?
96

Chapter 5. Joint Probability Distributions and Random Samples
STAT 155
Exercise 5.2When an automobile is stopped by a roving safety patrol, eachtire is checked for tire wear, and each headlight is checked to see whetherit is properly aimed.LetXdenote the number of headlights that needadjustment, and letYdenote the number of defective tires.(a) IfXandYare independent withpX(0) =.5,pX(1) =.3,pX(2) =.2,andpY(0) =.6,pY(1) =.1,pY(2) =pY(3) =.05,pY(4) =.2, display thejoint pmf of(X, Y)in a joint probability table.(b) ComputeP(X1andY1)from the joint probability table, andverify that it equals the productP(X1)·P(Y1).(c) What isP(X+Y= 0), the probability of no violations?(d) ComputeP(X+Y1).97

Chapter 5. Joint Probability Distributions and Random Samples
STAT 155
Two Continuous Random Variables
Let
X
and
Y
be continuous random variables. A
joint probability
density function
f
(
x, y
)
for these two variables is a function
satisfying
f
(
x, y
)
≥
0
and
R
1
-1
R
1
-1
f
(
x, y
)
dx dy
= 1
.