Page 244 - Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 227

Assuming independence, we have
y†; 7:88†
F Y n y†ˆ F X 1 y†F X 2 y† F X n
(y) ˆ F X (y), the result is
and, if each F X j
n
y†ˆ ‰F X y†Š : 7:89†
F Y n
The pdf of Y n can be easily derived from the above. When the X j are contin-
uous, it has the form
y†
dF Y n n 1
f y†ˆ ˆ n‰F X y†Š f y†: 7:90†
dy
Y n X
The PDF of Z n is determined in a similar fashion. In this case,

z†ˆ P Z n z†ˆ P at least one X j z†
F Z n
ˆ P X 1 z [ X 2 z [ [ X n z†
ˆ 1 P X 1 > z \ X 2 > z \ \ X n > z†:

When the X j are independent and identically distributed, the foregoing gives

z†Š
F Z n z†ˆ 1 ‰1 F X 1 z†Š‰1 F X 2 z†Š ‰1 F X n
n 7:91†
ˆ 1 ‰1 F X z†Š :
If random variables X j are continuous, the pdf of Z n is

f z†ˆ n‰1 F X z†Š n 1 f z†: 7:92†
X
Z n
(y) and
The next step in our development is to determine the forms of F Y n
(z) as expressed by Equations (7.89) and (7.91) as n !1. Since the initial
F Z n
distribution F X (x) of each X j is sometimes unavailable, we wish to examine
(y) and
whether Equations (7.89) and (7.91) lead to unique distributions for F Y n
(z), respectively, independent of the form of F X (x). This is not unlike
F Z n
looking for results similar to the powerful ones we obtained for the normal
and lognormal distributions via the central limit theorem.
(z) become increasingly
Although the distribution functions F Y n (y) and F Z n
insensitive to exact distributional features of X j as n !1, 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,
(z) are classified into three types based
the asymptotic forms of F Y n (y) and F Z n
on general features in the distribution tails of X j . Type I is sometimes referred

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