Soong T.T. Fundamentals of Probability and Statistics for Engineers
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[see Equation (2.26)], for three events A, B, and C, we have, in the case of three
random variables X, Y, and Z,
Hence, for the general case of n random variables, X
1
,X
2
,...,X
n
,or X,wecan
write
In the event that these random variables are mutually independent, Equations
(3.49) become
Example 3.9. To show that joint mass functions are sometimes more easily
found by finding first the conditional mass functions, let us consider a traffic
problem as described below.
Problem: a group of n cars enters an intersection from the south. Through
prior observations, it is estimated that each car has the probability p of turning
east, probability q of turning west, and probability r of going straight on
(p q r 1). Assume that drivers behave independently and let X be the
number of cars turning east and Y the number turning west. Determine the
jpmf p
XY
(x, y).
Answer: since
we proceed by determining p
XY
(x y) and p
Y
(y). The marginal mass function
p
Y
(y) is found in a way very similar to that in the random walk situation
described in Example 3.5. Each car has two alternatives: turning west, and
not turning west. By enumeration, we can show that it has a binomial distribu-
tion (to be more fully justified in Chapter 6)
64
Fundamentals of Probability and Statistics for Engineers
p
XYZ
x; y; zp
XYZ
xjy; zp
YZ
yjzp
Z
z
f
XYZ
x; y; zf
XYZ
xjy; zf
YZ
yjzf
Z
z
9
=
;
3:48
p
X
xp
X
1
X
2
...X
n
x
1
jx
2
;...; x
n
p
X
2
...X
n
x
2
jx
3
;...; x
n
...p
X
n1
X
n
x
n1
jx
n
p
X
n
x
n
;
f
X
xf
X
1
X
2
...X
n
x
1
jx
2
;...;x
n
f
X
2
...X
n
x
2
jx
3
;...;x
n
...f
X
n1
X
n
x
n1
jx
n
f
X
n
x
n
:
9
=
;
3:49
p
X
xp
X
1
x
1
p
X
2
x
2
...p
X
n
x
n
;
f
X
xf
X
1
x
1
f
X
2
x
2
...f
X
n
x
n
:
)
3:50
p
XY
x; yp
XY
xjyp
Y
y;
j
p
Y
y
n
y
q
y
1 q
ny
; y 1; 2; ...: 3:51
TLFeBOOK
Consider now the conditional mass function p
XY
(x y). With Y y having
happened, the situation is again similar to that for determining p
Y
(y) except
that the number of cars available for taking possible eastward turns is now
n y; also, here, the probabilities p and r need to be renormalized so that they
sum to 1. Hence, p
XY
(x y) takes the form
Finally, we have p
XY
(x,y) as the product of the two expressions given by
Equations (3.51) and (3.52). The ranges of values for x and y are x 0,1,...,
n y,and y 0,1,...,n.
Note that p
XY
(x,y) has a rather complicated expression that could not have
been derived easily in a direct way. This also points out the need to exercise care
in determining the limits of validity for x and y.
Ex ample 3. 10. Problem: resistors are designed to have a resistance R of
50 2 . Owing to imprecision in the manufacturing process, the actual density
function of R has the form shown by the solid curve in Figure 3.18. Determine
the density function of R after screening – that is, after all the resistors having
resistances beyond the 48–52 range are rejected.
Answer: we are interested in the conditional density function, f
R
(r A), where
A is the event . This is not the usual conditional density function
but it can be found from the basic definition of conditional probability.
We start by considering
48 50
52
f
R
f
R
(
r\A
)
f
R
(
r
)
r
Figure 3. 18 The actual, f
R
(r), and conditional, f
R
(r A), for Example 3.10
Random Variables and Probability Distributions
65
j
j
p
XY
xjy
n y
x
p
r p
x
1
p
r p
nyx
; x 0; 1;...; n y; y 0; 1;...; n:
3:52
j
f48 R 52g
F
R
rjAPR rj48 R 52
PR r \ 48 R 52
P48 R 52
:
j
(Ω)
TLFeBOOK
However,
Hence,
where
is a constant.
The desired f
R
(r A) is then obtained from the above by differentiation. We
obtain
It can be seen from Figure 3.18 (dashed line) that the effect of screening is
essentially a truncation of the tails of the distribution beyond the allowable
limits. This is accompanied by an adjustment within the limits by a multi-
plicative factor 1/c so that the area under the curve is again equal to 1.
FU RTHER READING AND COMMENTS
We discussed in Section 3.3 the determination of (unique) marginal distributions from a
knowledge of joint distributions. It should be noted here that the knowledge of marginal
distributionsdoes not in generallead to a uniquejoint distribution.Thefollowing reference
shows that all joint distributions having a specified set of marginals can be obtained by
repeated applications of the so-called transformation to the product of the marginals:
Becker, P.W., 1970, ‘‘A Note on Joint Densities which have the Same Set of Marginal
Densities’’, in Proc. International Symp. Information Theory, Elsevier Scientific Pub-
lishers, The Netherlands.
66 Fundamentals of Probability and Statistics for Engineers
R r \ 48 R 52
;; for r < 48;
48 R r; for 48 r 52;
48 R 52; for r > 52:
8
>
<
>
:
F
R
rjA
0; for r < 48;
P48 R r
P48 R 52
Z
r
48
f
R
rdr
c
; for 48 r 52;
1; for r > 52;
8
>
>
>
>
>
<
>
>
>
>
>
:
c
Z
52
48
f
R
rdr:
j
f
R
rjA
dF
R
rjA
dr
f
R
r
c
; for 48 r 52
0; elsewhere
8
<
:
TLFeBOOK
PROBLEMS
3.1 For each of the functions given below, determine constant a so that it possesses all
the properties of a probability distribution function (PDF). Determine, in each case,
its associated probability density function (pdf ) or probability mass function (pmf)
if it exists and sketch all functions.
(a) Case 1:
(b) Case 2:
(c) Case 3:
(d) Case 4:
(e) Case 5:
(f) Case 6:
(g) Case 7:
3.2 For each part of Problem 3.1, determine:
(a) P(X 6);
(b) P(
1
2
< X 7).
Random Variables and Probability Distributions
67
Fx
0; for x < 5;
a; for x 5:
Fx
0; for x < 5;
1
3
; for 5 x < 7;
a; for x 7:
8
>
<
>
:
Fx
0; for x < 1;
X
k
j1
1=a
j
; for k x < k 1; and k 1; 2; 3; ...:
8
>
<
>
:
Fx
0; for x 0;
1 e
ax
; for x > 0:
Fx
0; for x < 0;
x
a
; for 0 x 1;
1; for x > 1:
8
<
:
Fx
0; for x < 0;
a sin
1
x
p
; for 0 x 1;
1; for x > 0:
8
<
:
Fx
0; for x < 0;
a1 e
x=2
1
2
; for x 0:
(
TLFeBOOK
3.3 For p
X
(x) and f
X
(x) in Figure 3.19(a) and 3.19(b) respectively, sketch roughly in
scale the corresponding PDF F
X
(x) and show on all graphs the procedure for
finding P(2 < X < 4).
3.4 For each part, find the corresponding PDF for random variable X.
(a) Case 1:
(b) Case 2:
(c) Case 3:
3.5 The pdf of X is shown in Figure 3.20.
(a) Determine the value of a.
(b) Graph F
X
(x) approximately.
(c) Determine P(X 2).
(d) Determine P(X 2 X 1).
3.6 The life X, in hours, of a certain kind of electronic component has a pdf given by
Determine the probability that a component will survive 150 hours of operation.
(a)
123
0.6
0.2
x
p
X
(
x
)
(b)
f
X
(
x
)
4
x
Figure 3. 19 The probability mass function, p
X
(x), and probability density function,
f
X
(x), for Problem 3.3
68 Fundamentals of Probability and Statistics for Engineers
f
X
x
0:1; for 90 x < 100;
0; elsewhere:
f
X
x
21 x; for 0 x < 1;
0; elsewhere:
f
X
x
1
1 x
2
; for 1 < x < 1:
j
f
X
x
0; for x < 100;
100
x
2
; for x 100:
8
<
:
TLFeBOOK
3.7 Let T denote the life (in months) of a light bulb and let
(a) Plot f
T
(t) against t.
(b) Derive F
T
(t) and plot F
T
(t) against t.
(c) Determine using f
T
(t), the probability that the light bulb will last at least 15
months.
(d) Determine, using F
T
(t), the probability that the light bulb will last at least 15
months.
(e) A light bulb has already lasted 15 months. What is the probability that it will
survive another month?
3.8 The time, in minutes, required for a student to travel from home to a morning
class is uniformly distributed between 20 and 25. If the student leaves home
promptly at 7:38 a.m., what is the probability that the student will not be late for
class at 8:00 a.m.?
3.9 In constructing the bridge shown in Figure 3.21, an engineer is concerned with
forces acting on the end supports caused by a randomly applied concentrated load
P, the term ‘randomly applied’meaning that the probability of the load lying in any
region is proportional only to the length of that region. Suppose that the bridge has
a span 2b. Determine the PDF and pdf of random variable X, which is the distance
from the load to the nearest edge support. Sketch these functions.
f
X
(
x
)
a
–3
3
x
Figure 3. 20 The probability density function, f
X
(x), for Problem 3.5
2
b
P
Figure 3.21 Diagram of the bridge, for Problem 3.9
Random Variables and Probability Distributions
69
f
T
t
1
15
t
450
; for 0 t 30;
0; elsewhere:
8
<
:
TLFeBOOK
3.10 Fire can erupt at random at any point along a stretch of forest AB. The fire
station is located as shown in Figure 3.22. Determine the PDF and pdf of
X, representing the distance between the fire and the fire station. Sketch these
functions.
3.11 Pollutant concentrations caused by a pollution source can be modeled by the pdf
(a > 0):
where R is the distance from the source. Determine the radius within which 95% of
the pollutant is contained.
3.12 As an example of a mixed probability distribution, consider the following problem:
a particle is at rest at the origin (x 0) at time t 0. At a randomly selected time
uniformly distributed over the interval 0 < t < 1, the particle is suddenly given a
velocity v in the positive x direction.
(a) Show that X, the particle position at t(0 < t < 1), has the PDF shown in Figure
3.23.
(b) Calculate the probability that the particle is at least v/3 away from the origin at
Fire station
a
A
B
a
d
b
Figure 3.22 Position of the fire station and stretch of forest, AB, for Problem 3.10
F
X
(
x
)
F
X
(
x
)=1–
t
+—
x
t
x
1
1–
t
Figure 3. 23 The probability distribution function, F
X
(x), for Problem 3.12
70 Fundamentals of Probability and Statistics for Engineers
f
R
r
0; for r < 0;
ae
ar
; for r 0;
t 1/2.
ν
ν
TLFeBOOK
3.13 For each of the joint probability mass functions (jpmf), p
XY
(x,y), or joint prob-
ability density functions (jpdf), f
XY
(x, y), given below (cases 1–4), determine:
(a) the marginal mass or density functions,
(b) whether the random variables are independent.
(i) Case 1
(ii) Case 2:
(iii) Case 3
(iv) Case 4
3.14 Suppose X and Y have jpmf
(a) Determine marginal pmfs of X and Y.
(b) Determine P(X 1).
(c) Determine P(2X Y).
3.15 Let X
1
,X
2
,andX
3
be independent random variables, each taking values 1 with
probabilities 1/2. Define random variables Y
1
,Y
2
,andY
3
by
Show that any two of these new random variables are independent but that Y
1
,Y
2
,
and Y
3
are not independent.
3.16 The random variables X and Y are distributed according to the jpdf given by
Case 2, in Problem 3.13(ii). Determine:
(a)
(b)
Random Variables and Probability Distributions
71
p
XY
x; y
0:5; for x; y1; 1;
0:1; for x; y1; 2;
0:1; for x; y2; 1;
0:3; for x; y2; 2:
8
>
>
>
<
>
>
>
:
f
XY
x; y
ax y; for 0 < x 1; and 1 < y 2;
0; elsewhere:
f
XY
x; y
e
xy
; for x; y > 0; 0;
0; elsewhere:
f
XY
x; y
4yx ye
xy
; for 0 < x < 1; and 0 < y x;
0; elsewhere:
p
XY
x; y
0:1; for x; y1; 1;
0:2; for x; y1; 2;
0:3; for x; y2; 1;
0:4; for x; y2; 2:
8
>
>
>
<
>
>
>
:
Y
1
X
1
X
2
; Y
2
X
1
X
3
; Y
3
X
2
X
3
P(X 0:5 \ Y > 1:0).
P(XY <
1
2
).
TLFeBOOK
3.17 Let random variable X denote the time of failure in years of a system for which the
PDF is F
X
(x). In terms of F
X
(x), determine the probability
which is the conditional distribution function of X given that the system did not fail
up to 100 years.
3.18 The pdf of random variable X is
DetermineP(X>b X<b/2)with 1<b<0.
3.19 Using the joint probability distribution given in Example 3.5 for random variables
X and Y , determine:
(a) P(X > 3).
(b) P(0 Y < 3).
(c) P(X > 3 Y 2).
3.20 Let
(a) What must be the value of k?
(b) Determine the marginal pdfs of X and Y .
(c) Are X and Y statistically independent? Why?
3.21 A commuter is accustomed to leaving home between 7:30 a.m and 8:00 a.m., the drive
to the station taking between 20 and 30 minutes. It is assumed that departure time and
travel time for the trip are independent random variables, uniformly distributed over
their respective intervals. There are two trains the commuter can take; the first leaves
at 8:05 a.m. and takes 30 minutes for the trip, and the second leaves at 8:25 a.m. and
takes 35 minutes. What is the probability that the commuter misses both trains?
3.22 The distance X (in miles) from a nuclear plant to the epicenter of potential earth-
quakes within 50 miles is distributed according to
and the magnitude Y of potential earthquakes of scales 5 to 9 is distributed
according to
72 Fundamentals of Probability and Statistics for Engineers
(c) P(X 0:5jY 1:5).
(d) P(X 0:5jY 1:5).
PX xjX 100;
f
X
x
3x
2
; for 1 < x 0;
0; elsewhere:
j
j
f
XY
x; y
ke
xy
; for 0 < x < 1; and 0 < y < 2;
0; elsewhere:
f
X
x
2x
2500
; for 0 x 50;
0; elsewhere;
8
<
:
f
Y
y
39 y
2
64
; for 5 y 9;
0; elsewhere:
8
<
:
TLFeBOOK
Assume that X and Y are independent. Determine P(X 25 Y > 8), the prob-
ability that the next earthquake within 50 miles will have a magnitude greater than
8 and that its epicenter will lie within 25 miles of the nuclear plant.
3.23 Let random variables X and Y be independent and uniformly distributed in the
square (0,0) < (X,Y ) < (1,1). Determine the probability that XY < 1/2.
3.24 In splashdown maneuvers, spacecrafts often miss the target because of guidance
inaccuracies, atmospheric disturbances, and other error sources. Taking the origin
of the coordinates as the designed point of impact, the X and Y coordinates of the
actual impact point are random, with marginal density functions
Assume that the random variables are independent. Show that the probability
of a splashdown lying within a circle of radius a centered at the origin
is 1
3.25 Let X
1
,X
2
,...,X
n
be independent and identically distributed random variables,
each with PDF F
X
(x). Show that
The above are examples of extreme-value distributions. They are of considerable
practical importance and will be discussed in Section 7.6.
3.26 In studies of social mobility, assume that social classes can be ordered from 1
(professional) to 7 (unskilled). Let random variable X
k
denote the class order of the
kth generation. Then, for a given region, the following information is given:
(i) The pmf of X
0
is described by
(ii) The conditional probabilities
and for
every k are given in Table 3.2.
Table 3. 2 for Problem 3.26
ij
1234567
1 0.388 0.107 0.035 0.021 0.009 0.000 0.000
2 0.146 0.267 0.101 0.039 0.024 0.013 0.008
3 0.202 0.227 0.188 0.112 0.075 0.041 0.036
4 0.062 0.120 0.191 0.212 0.123 0.088 0.083
5 0.140 0.206 0.357 0.430 0.473 0.391 0.364
6 0.047 0.053 0.067 0.124 0.171 0.312 0.235
7 0.015 0.020 0.061 0.062 0.125 0.155 0.274
Random Variables and Probability Distributions
73
\
f
X
x
1
2
1=2
e
x
2
=2
2
; 1 < x < 1;
f
Y
y
1
2
1=2
e
y
2
=2
2
; 1 < y < 1:
e
a
2
/2
2
.
PminX
1
; X
2
; ...; X
n
u1 1 F
X
u
n
;
PmaxX
1
; X
2
; ...; X
n
uF
X
u
n
:
p
X
0
(1) 0:00, p
X
0
(2) 0:00, p
X
0
(3) 0:04,
p
X
0
(4) 0:06, p
X
0
(5) 0:11, p
X
0
(6) 0:28, and p
X
0
(7) 0:51.
P(X
k1
ijX
k
j) for i, j 1, 2, ..., 7
P(X
k1
ijX
k
j)
TLFeBOOK