Soong T.T. Fundamentals of Probability and Statistics for Engineers
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I
x
( , ). If we write F
X
(x) with parameters and in the form F( ), the
correspondence between I
x
( , ) and F( ) is determined as follows. If
,then
If then
Another method of evaluating F
X
(x) in Equation (7.74) is to note the
similarity in form between f
X
(x) and p
Y
(k) of a binomial random variable Y
for the case where and are positive integers. We see from Equation
(6.2) that
Also, f
X
(x) in Equation (7.70) with and being positive integers takes the
form
and we easily establish the relationship
where p
Y
(k) is evaluated at with and For
example, the value of f
X
(0 5) with and is numerically equal to
2p
Y
(1) with n 1, and p 0 5; here p
Y
(1) can be found from Equation (7.77)
or from Table A.1 for binomial random variables.
Similarly, the relationship between F
X
(x) and F
Y
(k) can be established. It
takes the form
with and The PDF F
Y
(y) for a binomial
random variable Y is also widely tabulated and it can be used to advantage
here for evaluating F
X
(x) associated with the beta distribution.
Example 7.8. Problem: in order to establish quality limits for a manufactured
item, 10 independent samples are taken at random and the quality limits are
224
Fundamentals of Probability and Statistics for Engineers
x
; ,
, x; ,
,
Fx; ; I
x
; : 7:75
<,
Fx; ; 1 I
1x
; : 7:76
p
Y
k
n!
k!n k!
p
k
1 p
nk
; k 0; 1; ...; n: 7:77
f
X
x
1!
1! 1!
x
1
1 x
1
;; 1; 2; ...; 0 x 1; 7:78
f
X
x 1p
Y
k;; 1; 2; ...; 0 x 1; 7:79
k 1, n 2, p x.
: 2, 1,
:
F
X
x1 F
Y
k;; 1; 2; ...; 0 x 1; 7:80
k 1, n 2,
p x.
TLFeBOOK
established by using the lowest and highest sample values. What is the prob-
ability that at least 50% of the manufactured items will fail within these limits?
Answer: let X be the proportion of items taking values within the established
limits. Its pdf thus takes the form of Equation (7.73), with n 10,r 1, and
Hence, and
The desired probability is
According to Equation (7.80), the value of F
X
(0 50) can be found from
where Y is binomial and k 1 8, n 2 9, and p 0 50.
Using Table A.1, we find that
Equations (7.81) and (7.82) yield
7.5.2 GENERALIZED BETA DISTRIBUTION
The beta distribution can be easily generalized from one restricted to unit
interval (0,1) to one covering an arbitrary interval (a,b). Let Y be such
a generalized beta random variable. It is clear that the desired transforma-
tion is
where X is beta-distributed according to Equation (7.70). Equation (7.85)
represents a monotonic transformation from X and Y and the procedure
Some Important Continuous Distributions 225
s
1.
10 1 1 1 9, 1 1 2,
f
X
x
11
92
x
8
1 x;
10!
8!
x
8
1 x; for 0 x 1;
0; elsewhere:
PX > 0:501 P X 0: 501 F
X
0:50: 7:81
:
F
X
0:501 F
Y
k; 7:82
:
F
Y
81 p
Y
91 0:002 0:998: 7:83
PX > 0:501 F
X
0:501 1 F
Y
80:998: 7:84
Y b aX a; 7:85
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developed in Chapter 5 can be applied to determine the pdf of Y in a straight-
forward manner. Following Equation (5.12), we have
7.6 EXTREME-VALUE DISTRIBUTIO NS
A structural engineer, concerned with the safety of a structure, is often inter-
ested in the maximum load and maximum stress in structural members. In
reliability studies, the distribution of the life of a system having n components
in series (where the system fails if any component fails) is a function of the
minimum time to failure of these components, whereas for a system with a
parallel arrangement (where the system fails when all components fail) it is
determined by the distribution of maximum time to failure. These examples
point to our frequent concern with distributions of maximum or minimum
values of a number of random variables.
To fix ideas, let X
j
,j 1,2,...,n, denote the jth gust velocity of n gusts
occurring in a year, and let Y
n
denote the annual maximum gust velocity. We
are interested in the probability distribution of Y
n
in terms of those of X
j
.Inthe
following development, attention is given to the case where random variables
X
j
,j 1,2,...,n, are independent and identically distributed with PDF F
X
(x)
and pdf f
X
(x) or pmf p
X
(x). Furthermore, asymptotic results for are
our primary concern. For the wind-gust example given above, these conditions
are not unreasonable in determining the distribution of annual maximum gust
velocity. We will also determine, under the same conditions, the minimum Z
n
of random variables X
1
,X
2
,..., and X
n
, which is also of interest in practical
applications.
The random variables Y
n
and Z
n
are defined by
The PDF of Y
n
is
226
Fundamentals of Probability and Statistics for Engineers
f
Y
y
1
b a
1
y a
1
b y
1
; for a x b;
0; elsewhere:
8
>
<
>
:
7:86
n !1
Y
n
maxX
1
; X
2
; ...; X
n
;
Z
n
minX
1
; X
2
; ...; X
n
:
7:87
F
Y
n
yPY
n
yPall X
j
y
PX
1
y \ X
2
y \\X
n
y:
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Assuming independence, we have
and, if each F
X
j
(y) F
X
(y), the result is
The pdf of Y
n
can be easily derived from the above. When the X
j
are contin-
uous, it has the form
The PDF of Z
n
is determined in a similar fashion. In this case,
When the X
j
are independent and identically distributed, the foregoing gives
If random variables X
j
are continuous, the pdf of Z
n
is
The next step in our development is to determine the forms of F
Y
n
(y) and
F
Z
n
(z) as expressed by Equations (7.89) and (7.91) as Since the initial
distribution F
X
(x) of each X
j
is sometimes unavailable, we wish to examine
whether Equations (7.89) and (7.91) lead to unique distributions for F
Y
n
(y) and
F
Z
n
(z), respectively, independent of the form of F
X
(x). This is not unlike
looking for results similar to the powerful ones we obtained for the normal
and lognormal distributions via the central limit theorem.
Although the distribution functions F
Y
n
(y) and F
Z
n
(z) become increasingly
insensitive to exact distributional features of X
j
as no unique results
can be obtained that are completely independent of the form of F
X
(x). Some
features of the distribution function F
X
(x) are important and, in what follows,
the asymptotic forms of F
Y
n
(y) and F
Z
n
(z) are classified into three types based
on general features in the distribution tails of X
j
. Type I is sometimes referred
Some Important Continuous Distributions 227
F
Y
n
yF
X
1
yF
X
2
yF
X
n
y; 7:88
F
Y
n
yF
X
y
n
:7:89
f
Y
n
y
dF
Y
n
y
dy
nF
X
y
n1
f
X
y:7:90
F
Z
n
zPZ
n
zPat least one X
j
z
PX
1
z [ X
2
z [[X
n
z
1 PX
1
> z \ X
2
> z \\X
n
> z:
F
Z
n
z1 1 F
X
1
z1 F
X
2
z1 F
X
n
z
1 1 F
X
z
n
:
7:91
f
Z
n
zn1 F
X
z
n1
f
X
z: 7:92
n !1.
n !1,
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to as Gumbel’s extreme value distribution, and included in Type III is the
important Weibull distribution.
7.6.1 TYPE-I ASYMPTOTIC DISTRIBUTIONS OF EXTREME
VALUES
Consider first the Type-I asymptotic distribution of maximum values. It is the
limiting distribution of Y
n
(as ) from an initial distribution F
X
(x) of
which the right tail is unbounded and is of an exponential type; that is, F
X
(x)
approaches 1 at least as fast as an exponential distribution. For this case, we
can express F
X
(x) in the form
where g(x) is an increasing function of x. A number of important distributions
fall into this category, such as the normal, lognormal, and gamma distributions.
Let
We have the following important result (Theorem 7.6).
Theorem 7. 6: let random variables X
1
,X
2
,..., and X
n
be independent and
identically distributed with the same PDF F
X
(x). If F
X
(x) is of the form given
by Equation (7.93), we have
where and u are two parameters of the distribution.
Proof of Theorem 7.6: we shall only sketch the proof here; see Gumbel (1958)
for a more comprehensive and rigorous treatment.
Let us first define a quantity u
n
, known as the characteristic value of Y
n
,by
It is thus the value of X
j
,j 1,2,...,n, at which P (X
j
u
n
) 1 1/n. As n
becomes large, F
X
(u
n
) approaches unity, or, u
n
is in the extreme right-hand tail
of the distribution. It can also be shown that u
n
is the mode of Y
n
, which can
be verified, in the case of X
j
being continuous, by taking the derivative of f
Y
n
(y)
in Equation (7.90) with respect to y and setting it to zero.
228
Fundamentals of Probability and Statistics for Engineers
n !1
F
X
x1 expgx; 7:93
lim
n!1
Y
n
Y: 7:94
F
Y
yexpfexpy u; 1 < y < 1; 7:95
(>0)
F
X
u
n
1
1
n
: 7:96
TLFeBOOK
If F
X
(x) takes the form given by Equation (7.93), we have
or
Now, consider F
Y
n
(y) defined by Equation (7.89). In view of Equation (7.93),
it takes the form
In the above, we have introduced into the equation the factor exp [g(u
n
)]/n,
which is unity, as shown by Equation (7.97).
Since u
n
is the mode or the ‘most likely’ value of Y
n
, function g(y) in
Equation (7.98) can be expanded in powers of (y u
n
) in the form
where
n
dg(y)/dy is evaluated at y u
n
. It is positive, as g(y) is an increasing
function of y. Retaining only up to the linear term in Equation (7.99) and
substituting it into Equation (7.98), we obtain
in which
n
and u
n
are functions only of n and not of y. Using the identity
for any real c, Equation (7.100) tends, as n , to
which was to be proved. In arriving at Equation (7.101), we have assumed that
as n , F
Y
n
(y) converges to F
Y
(y) as Y
n
converges to Y in some probabilistic
sense.
Some Important Continuous Distributions 229
1 expgu
n
1
1
n
;
expgu
n
n
1: 7:97
F
Y
n
yf1 expgyg
n
1
expgu
n
expgy
n
n
1
expfgygu
n
g
n
n
:
7:98
gygu
n
n
yu
n
;7:99
F
Y
n
y 1
exp
n
y u
n
n
n
; 7:100
lim
n!1
1
c
n
n
expc;
!1
F
Y
yexpfexpy ug; 7:101
!1
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The mean and variance associated with the Type-I maximum-value distribu-
tion can be obtained through integration using Equation (7.90). We have noted
that u is the mode of the distribution, that is, the value of y at which f
Y
(y) is
maximum. The mean of Y is
where
It is seen from the above that u and are, respectively, the location and scale
parameters of the distribution. It is interesting to note that the skewness
coefficient, defined by Equation (4.11), in this case is
which is independent of and u. This result indicates that the Type-I
maximum-value distribution has a fixed shape with a dominant tail to the right.
A typical shape for f
Y
(y) is shown in Figure 7.14.
The Type-I asymptotic distribution for minimum values is the limiting
distribution of Z
n
in Equation (7.91) as n from an initial distribution
F
X
(x) of which the left tail is unbounded and is of exponential type as it decreases
to zero on the left. An example of F
X
(x) that belongs to this class is the normal
distribution.
The distribution of Z
n
as n can be derived by means of procedures
given above for Y
n
through use of a symmetrical argument. Without giving
details, if we let
y
f
Y
(
y
)
Figure 7.14 Typical plot of a Type-I maximum-value distribution
230 Fundamentals of Probability and Statistics for Engineers
0 577 is Euler’s constant; and the variance is given by
m
Y
u
;7:102
'
2
Y
2
6
2
: 7:103
1
' 1:1396;
!1
!1
lim
n!1
Z
n
Z; 7:104
:
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the PDF of Z can be shown to have the form
where and u are again the two parameters of the distribution.
It is seen that Type-I asymptotic distributions for maximum and minimum
values are mirror images of each other. The mode of Z is u, and its mean,
variance, and skewness coefficients are, respectively,
For probability calculations, values for probability distribution functions
F
Y
(y) and F
Z
(z) over various ranges of y and z are available in, for example,
Microsoft Excel 2000 (see Appendix B).
Ex ample 7. 9. Problem: the maximum daily gasoline demand Y during the
month of May at a given locality follows the Type-I asymptotic maximum-
value distribution, with m
Y
2and
Y
1, measured in thousands of gallons.
Determine (a) the probability that the demand will exceed 4000 gallons in
any day during the month of May, and (b) the daily supply level that for 95%
of the time will not be exceeded by demand in any given day.
Answer: it follows from Equations (7.102) and (7.103) that parameters and
u are determined from
For part (a), the solution is
For part (b), we need to determine y such that
Some Important Continuous Distributions 231
F
Z
z1expfexpz ug; 1 < z < 17:105
m
Z
u
2
Z
2
6
2
1
'1:1396
9
>
>
>
>
=
>
>
>
>
;
7:106
6
p
Y
6
p
1:282;
u m
Y
0:577
2
0:577
1:282
1:55:
PY > 41 F
Y
4
1 expfexp1:2824 1:55g
1 0:958 0:042:
F
Y
yPY y0:95;
TLFeBOOK
or
Taking logarithms of Equation (7.107) twice, we obtain
that is, the required supply level is 3867 gallons.
Example 7.10. Problem: consider the problem of estimating floods in the
design of dams. Let y
T
denote the maximum flood associated with return
period T. Determine the relationship between y
T
and T if the maximum river
flow follows the Type-I maximum-value distribution. Recall from Example 6.7
(page 169) that the return period T is defined as the average number of years
between floods for which the magnitude is greater than y
T
.
Answer: assuming that floods occur independently, the number of years
between floods with magnitudes greater than y
T
assumes a geometric distribu-
tion. Thus
Now, from Equation (7.101),
where b (y
T
u). The substitution of Equation (7.109) into Equation
(7.108) gives the required relationship.
For values of y
T
where F
Y
(y
T
) 1, an approximation can be made by
noting from Equation (7.109) that
Since F
Y
(y
T
) is close to 1, we retain only the first term in the foregoing
expansion and obtain
Equation (7.108) thus gives the approximate relationship
232
Fundamentals of Probability and Statistics for Engineers
expfexp1:282y 1:55g 0:95: 7:107
y 3:867;
T
1
PY > y
T
1
1 F
Y
y
T
: 7:108
F
Y
y
T
expexpb; 7:109
!
expbln F
Y
y
T
fF
Y
y
T
1
1
2
F
Y
y
T
1
2
g:
1 F
Y
y
T
'expb:
y
T
u 1
1
u
ln T
; 7:110
TLFeBOOK
where u is the scale factor and the value of u describes the characteristics of
a river; it varies from 1.5 for violent rivers to 10 for stable or mild rivers.
In closing, let us remark again that the Type-I maximum-value distribution
is valid for initial distributions of such practical importance as normal, lognor-
mal, and gamma distributions. It thus has wide applicability and is sometimes
simply called the extreme value distribution.
7.6.2 TYPE-II ASYMPTOTIC DISTRIBUTIONS OF EXTREME
VALUES
The Type-II asymptotic distribution of maximum values arises as the limiting
distribution of Y
n
as n from an initial distribution of the Pareto type, that
is, the PDF F
X
(x) of each X
j
is limited on the left at zero and its right tail is
unbounded and approaches one according to
For this class, the asymptotic distribution of Y
n
,F
Y
(y), as n takes the
form
Let us note that, with F
X
(x) given by Equation (7.111), each X
j
has moments
only up to order r, where r is the largest integer less than k. If k > 1, the mean of
Y is
and, if k > 2, the variance has the form
The derivation of F
Y
(y) given by Equation (7.112) follows in broad outline
that given for the Type-I maximum-value asymptotic distribution and will not
be presented here. It has been used as a model in meteorology and hydrology
(Gumbel, 1958).
A close relationship exists between the Type-I and Type-II asymptotic
maximum-value distributions. Let Y
I
and Y
II
denote, respectively, these random
Some Important Continuous Distributions 233
!1
F
X
x1 ax
k
; a; k > 0; x 0: 7:111
!1
F
Y
yexp
y
v
k
; v; k > 0; y 0: 7:112
m
Y
v 1
1
k
; 7:113
2
Y
v
2
1
2
k
2
1
1
k
: 7:114
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