Fundamentals of Probability and Statistics for Engineers

Assuming independence, we have

and, if each FXj(y) FX(y), the result is

The pdf of Yn can be easily derived from the above. When the Xj are contin-
uous, it has the form

The PDF of Zn is de te rmined in a similar fashion. In this case,

When the Xj are independent and identically distributed, the foregoing gives

If random variables Xj are continuous, the pdf of Zn is

The next step in our development is to determine the forms of FYn(y) and
FZn (z) as expressed by Equations (7.89) and (7.91) as Since the initial
distribution FX(x) of each Xj is sometimes unavailable, we wish to examine
whether Equations (7.89) and (7.91) lead to unique distributions for FYn(y) and
FZn (z), respectively, independent of the form of FX(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 FYn(y) and FZn (z) become increasingly
insensitive to exact distributional features of Xj as no unique results
can be obtained that are co mpletely independent of the form of FX(x). Some
features of the distribution function FX(x) are important and, in what follows,
the asymptotic forms of FYn(y) and FZn (z) are classified into three types based
on general features in the distribution tails of Xj. Type I is sometimes referred

Some Important Continuous Distributions 227

FYn…y†ˆFX 1 …y†FX 2 …y†FXn…y†; … 7 : 88 †

ˆ

FYn…y†ˆ‰FX…y†Šn:… 7 : 89 †

fYn…y†ˆ

dFYn…y†

dy

ˆn‰FX…y†Šn^1 fX…y†:… 7 : 90 †

FZn…z†ˆP…Znz†ˆP…at least oneXjz†

ˆP…X 1 z[X 2 z[[Xnz†

ˆ 1 P…X 1 >z\X 2 >z\\Xn>z†:

FZn…z†ˆ 1 ‰ 1 FX 1 …z†Š‰ 1 FX 2 …z†Š‰ 1 FXn…z†Š

ˆ 1 ‰ 1 FX…z†Šn:

… 7 : 91 †

fZn…z†ˆn‰ 1 FX…z†Šn^1 fX…z†: … 7 : 92 †

n!1.

n!1,