Fundamentals of Probability and Statistics for Engineers

If FX(x) takes the form given by Equation (7.93), we have

or

Now, consider FYn(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(un)]/n,
which is unity, as shown by Equation (7.97).
Since un is the mode or the ‘most likely’ value of Yn, function g(y) in
Equation (7.98) can be expanded in powers of (y un) in the form

where n dg(y)/ dy is evaluated at y un. 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 un are functions only of n and not of y. Using the id entity

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 , FYn (y) converges to FY (y) as Yn converges to Y in some probabilistic
sense.

Some Important Continuous Distributions 229

1 exp‰g…un†Šˆ 1 

1

n

;

exp‰g…un†Š

n

ˆ 1 : … 7 : 97 †

FYn…y†ˆf 1 exp‰g…y†Šgn

ˆ 1 

exp‰g…un†Šexp‰g…y†Š

n

n

ˆ 1 

expf‰g…y†g…un†Šg

n

n

:

… 7 : 98 †



g…y†ˆg…un†‡ (^) n…yun†‡;… 7 : 99 †
ˆ ˆ
FYn…y†ˆ 1 
exp‰ (^) n…yun†Š
n
n
; … 7 : 100 †
lim
n!1

1 

c

n

n

ˆexp…c†;

!1

FY…y†ˆexpfexp‰ …yu†Šg; … 7 : 101 †

!1