Soong T.T. Fundamentals of Probability and Statistics for Engineers
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and
These are marginal pmfs of X and Y .
The joint probability distribution function F
XY
(x,y) can also be constructed,
by using Equation (3.22). Rather than showing it in three-dimensional form,
Figure 3.12 gives this function by indicating its value in each of the dividing
regions. One should also note that the arrays of indicated numbers beyond
y 5 are values associated with the marginal distribution function F
X
(x).
Similarly, F
Y
(y) takes those values situated beyond x 5. These observations
are also indicated on the graph.
The knowledge of the joint probability mass function permits us to make all
probability calculations of interest. The probability of any event being realized
involving X and Y is found by determining the pairs of values of X and Y that
give rise to this event and then simply summing over the values of p
XY
(x, y) for
all such pairs. In Example 3.5, suppose we wish to determine the probability of
X > Y ; it is given by
F
X
(
x
)
F
XY
(
x,y
)
F
Y
(
y
)
x
y
1
1
0.07776
0.33696
0.68256
0.91296
0.98976
35
1
3
0.01024
0.08920
0.31744
0.66304
0.92224
5
0.2592
0.6048
0.8352
0.9120
0.3456
0.5760
0.2304
0.0768
0.3072
0.6528
Zero
Zero
Zero
Figure 3. 12 The joint probability distribution function, F
XY
(x,y), for Example 3.5,
with p and q
54 Fundamentals of Probability and Statistics for Engineers
p
Y
y
X
i
p
XY
x
i
; y
p
5
; for y 0;
5p
4
q; for y 1;
10p
3
q
2
; for y 2;
10p
2
q
3
; for y 3;
5pq
4
; for y 4;
q
5
; for y 5:
8
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
:
0:4 0:6
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Example 3.6. Let us discuss again Example 2.11 in the context of random
variables. Let X be the random variable representing precipitation levels, with
values 1, 2, and 3 indicating low, medium, and high, respectively. The random
variable Y will be used for the peak flow rate, with the value 1 when it is critical
and 2 when noncritical. The information given in Example 2.11 defines jpmf
p
XY
(x,y), the values of which are tabulated in Table 3.1.
In order to determine the probability of reaching the critical level of peak
flow rate, for example, we simply sum over all p
XY
(x, y) satisfying y 1,
regardless of x values. Hence, we have
The definition of jpmf for more than two random variables is a direct extension
of that for the two-random-variable case. Consider n random variables
X
1
,X
2
,...,X
n
. Their jpmf is defined by
which is the probability of the intersection of n events. Its properties and
utilities follow directly from our discussion in the two-random-variable case.
Again, a more compact form for the jpmf is p
X
(x)whereX is an n-dimensional
random vector with components X
1
,X
2
,...,X
n
.
3.3.3 JOINT PROBABILITY DENSITY FUNCTION
As in the case of single random variables, probability density functions become
appropriate when the random variables are continuous. The joint probability
Table 3.1 Joint probability mass function for low, medium, and high precipitation
levels (x 1, 2, and 3, respectively) and critical and noncritical peak flow rates (y 1
and 2, respectively), for Example 3.6
yx
12 3
1 0.0 0.06 0.12
2 0.5 0.24 0.08
Random Variables and Probability Distributions
55
PX > YPX 5 \ Y 0PX 4 \ Y 1PX 3 \ Y 2
0:01024 0:0768 0:2304 0:31744:
PY 1p
XY
1; 1p
XY
2; 1p
XY
3; 10:0 0:06 0:12 0:18:
p
X
1
X
2
...X
n
x
1
; x
2
; ...; x
n
PX
1
x
1
\ X
2
x
2
\ ...\ X
n
x
n
;
3:23
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density function (jpdf) of two random variables, X and Y , is defined by the
partial derivative
Since F
XY
(x,y) is monotone nondecreasing in both x and y,f
XY
(x, y) is
nonnegative for all x and y. We also see from Equation (3.24) that
Moreover, with x
1
<x
2
,andy
1
<y
2
,
The jpdf f
XY
(x, y) defines a surface over the (x, y) plane. As indicated by
Equation (3.26), the probability that random variables X and Y fall within a
certain region R is equal to the volume under the surface of f
XY
(x, y) and
bounded by that region. This is illustrated in F igure 3.13.
f
XY
(
x, y
)
x
y
R
Figure 3. 13 A joint probability density function, f
XY
(x,y)
56 Fundamentals of Probability and Statistics for Engineers
f
XY
x; y
q
2
F
XY
x; y
qxqy
:
3:24
F
XY
x; yPX x \ Y y
Z
y
1
Z
x
1
f
XY
u; vdudv:
3:25
Px
1
< X x
2
\ y
1
< Y y
2
Z
y
2
y
1
Z
x
2
x
1
f
XY
x; ydxdy: 3:26
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We also note the following important properties:
Equation (3.27) follows from Equation (3.25) by letting and
this shows that the total volume under the f
XY
(x,y) surface is unity. To give
a derivation of Equation (3.28), we know that
Differentiating the above with respect to x gives the desired result immediately.
The density functions f
X
(x) and f
Y
(y) in Equations (3.28) and (3.29) are now
called the marginal density functions of X and Y , respectively.
Example 3.7. Problem: a boy and a girl plan to meet at a certain place between
9 a.m. and 10 a.m., each not waiting more than 10 minutes for the other. If all
times of arrival within the hour are equally likely for each person, and if their
times of arrival are independent, find the probability that they will meet.
Answer: for a single continuous random variable X that takes all values over
an interval a to b with equal likelihood, the distribution is called a uniform
distribution and its density function f
X
(x) has the form
The height of f
X
(x) over the interval (a, b) must be 1/(b a) in order that the
area is 1 below the curve (see Figure 3.14). For a two-dimensional case as
described in this example, the joint density function of two independent uni-
formly distributed random variables is a flat surface within prescribed bounds.
The volume under the surface is unity.
Let the boy arrive at X minutes past 9 a.m. and the girl arrive at Y minutes past
9 a.m. The jpdf f
XY
(x, y) thus takes the form shown in Figure 3.15 and is given by
Random Variables and Probability Distributions 57
Z
1
1
Z
1
1
f
XY
x; ydxdy 1; 3:27
Z
1
1
f
XY
x;ydyf
X
x;3:28
Z
1
1
f
XY
x;ydxf
Y
y:3:29
x, y !1, 1,
F
X
xF
XY
x; 1
Z
1
1
Z
x
1
f
XY
u; ydudy:
f
X
x
1
b a
; for a x b;
0; elsewhere:
8
<
:
3:30
f
XY
x; y
1
3600
; for 0 x 60; and 0 y 60;
0; elsewhere:
8
<
:
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The probability we are seeking is thus the volume under this surface over an
appropriate region R. For this problem, the region R is given by
and is shown in Figure 3.16 in the (x,y) plane.
The volume of interest can be found by inspection in this simple case.
Dividing R into three regions as shown, we have
P(they will meet) P X Y 10
ab
x
f
X
(
x
)
1
b
–
a
Figure 3. 14 A uniform density function, f
X
(x)
f
XY
(
x
,
y
)
y
060
60
3600
x
1
Figure 3. 15 The joint probability density function f
XY
(x,y), for Example 3.7
58 Fundamentals of Probability and Statistics for Engineers
R : jX Yj10
j j
251010
2
p
50
2
p
=3600
11
36
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Note that, for a more complicated jpdf, one needs to carry out the volume
integral
for volume calculations.
As an exercise, let us determine the joint probability distribution function
and the marginal density functions of random variables X and Y defined in
Example 3.7.
The JPDF of X and Y is obtained from Equation (3.25). It is clear that
Within the region (0, 0) (x, y) (60, 60), we have
For marginal density functions, Equations (3.28) and (3.29) give us
y
x
60
60
010
R
10
y
–
x
=10
x
–
y
=10
Figure 3. 16 Region R in Example 3.7
Random Variables and Probability Distributions
59
RR
R
f
XY
(x, y)dxdy
F
XY
x; y
0; for x; y < 0; 0;
1; for x; y > 60; 60:
(
F
XY
x; y
Z
y
0
Z
x
0
1
3600
dxdy
xy
3600
:
f
X
x
Z
60
0
1
3600
dy
1
60
; for 0 x 60;
0; elsewhere:
8
>
<
>
:
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Similarly,
Both random variables are thus uniformly distributed over the interval (0, 60).
Ex ample 3. 8. In structural reliability studies, the resistance Y of a structural
element and the force X applied to it are generally regarded as random vari-
ables. The probability of failure, p
f
, is defined by P(Y X). Suppose that the
jpdf of X and Y is specified to be
where a and b are known positive constants, we wish to determine p
f
.
The probability p
f
is determined from
where R is the region satisfying Y X. Since X and Y take only positive values,
the region R is that shown in Figure 3.17. Hence,
R
x
=
y
x
y
Figure 3. 17 Region R in Example 3.8
60 Fundamentals of Probability and Statistics for Engineers
f
Y
y
1
60
; for 0 y 60;
0; elsewhere:
8
<
:
f
XY
x; y
abe
axby
; for x; y > 0;
0; for x; y0;
(
p
f
Z
R
Z
f
XY
x; ydxdy;
p
f
Z
1
0
Z
1
y
abe
axby
dxdy
b
a b
:
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In closing this section, let us note that generalization to the case of many
random variables is again straightforward. The joint distribution function of n
random variables X
1
,X
2
,...,X
n
,or X, is given, by Equation (3.19), as
The corresponding joint density function, denoted by f
X
( x), is then
if the indicated partial derivatives exist. Various properties possessed by these
functions can be readily inferred from those indicated for the two-random-
variable case.
3.4 CONDITIONAL DISTRIBUTION AND INDEPENDENCE
The important concepts of conditional probability and independence intro-
duced in Sections 2.2 and 2.4 play equally important roles in the context of
random variables. The conditional distribution function of a random variable X,
given that another random variable Y has taken a value y, is defined by
Similarly, when random variable X is discrete, the definition of conditional mass
function of X given Y y is
Using the definition of conditional probability given by Equation (2.24),
we have
or
Random Variables and Probability Distributions 61
F
X
xPX
1
x
1
\ X
2
x
2
...\ X
n
x
n
: 3:31
f
X
x
q
n
F
X
x
qx
1
qx
2
...qx
n
;
3:32
F
XY
xjyPX xjY y:
3:33
p
XY
xjyPX xjY y:
3:34
p
XY
xjyPX xjY y
PX x \ Y y
PY y
;
p
XY
xjy
p
XY
x; y
p
Y
y
; if p
Y
y 6 0; 3:35
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which is expected. It gives the relationship between the joint jpmf and the
conditional mass function. As we will see in Example 3.9, it is sometimes more
convenient to derive joint mass functions by using Equation (3.35), as condi-
tional mass functions are more readily available.
If random variables X and Y are independent, then the definition of inde-
pendence, Equation (2.16), implies
and Equation (3.35) becomes
Thus, when, and only when, random variables X and Y are independent, their
jpmf is the product of the marginal mass functions.
Let X be a continuous random variable. A consistent definition of the
conditional density function of X given Y
is the derivative of
its corresponding conditional distribution function. Hence,
where F
XY
(x y) is defined in Equation (3.33). To see what this definition leads
to, let us consider
In terms of jpdf f
XY
(x, y), it is given by
By setting
nd and by taking the limit
Equation (3.40) reduces to
provided that f
Y
(y) 0.
62
Fundamentals of Probability and Statistics for Engineers
p
XY
xjyp
X
x; 3:36
p
XY
x; yp
X
xp
Y
y: 3:37
y, f
XY
(xjy),
f
XY
xjy
dF
XY
xjy
dx
; 3:38
j
Px
1
< X x
2
jy
1
< Y y
2
Px
1
< X x
2
\ y
1
< Y y
2
Py
1
< Y y
2
: 3:39
Px
1
< X x
2
jy
1
< Y y
2
Z
y
2
y
1
Z
x
2
x
1
f
XY
x;ydxdy
Z
y
2
y
1
Z
1
1
f
XY
x;ydxdy
Z
y
2
y
1
Z
x
2
x
1
f
XY
x;ydxdy
Z
y
2
y
1
f
Y
ydy: 3:40
x
1
1, x
2
x, y
1
y,a
y
2
y y,
y ! 0,
F
XY
xjy
Z
x
1
f
XY
u; ydu
f
Y
y
; 3:41
6
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Now we see that Equation (3.38) leads to
which is in a form identical to that of Equation (3.35) for the mass functions – a
satisfying result. We should add here that this relationship between the condi-
tional density function and the joint density function is obtained at the expense
of Equation (3.33) for F
XY
(x y). We say ‘at the expense of’ because the defin-
ition given to F
XY
(x y) does not lead to a convenient relationship between
F
XY
(x y) and F
XY
(x, y), that is,
This inconvenience, however, is not a severe penalty as we deal with density
functions and mass functions more often.
When random variables X and Y are independent, F
XY
(x y) F
X
(x) and, as
seen from Equation (3.42),
and
which shows again that the joint density function is equal to the product of the
associated marginal density functions when X and Y are independent.
Finally, let us note that, when random variables X and Y are discrete,
and, in the case of a continuous random variable,
Comparison of these equations with Equations (3.7) and (3.12) reveals they are
identical to those relating these functions for X alone.
Extensions of the above results to the case of more than two random vari-
ables are again straightforward. Starting from
Random Variables and Probability Distributions 63
f
XY
xjy
dF
XY
xjy
dx
f
XY
x; y
f
Y
y
; f
Y
y 6 0; 3:42
j
j
j
F
XY
xjy 6
F
XY
x; y
F
Y
y
: 3:43
j
f
XY
xjyf
X
x; 3:44
f
XY
x; yf
X
xf
Y
y; 3:45
F
XY
xjy
X
i : x
i
x
i1
p
XY
x
i
jy; 3:46
F
XY
xjy
Z
x
1
f
XY
ujydu: 3:47
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