Page 246 - Fundamentals of Probability and Statistics for Engineers

Some Important Continuous Distributions 229

If F X (x) takes the form given by Equation (7.93), we have
1
1 exp‰ g u n †Š ˆ 1 ;
n
or
exp‰g u n †Š
ˆ 1: 7:97†
n
(y) defined by Equation (7.89). In view of Equation (7.93),
Now, consider F Y n
it takes the form
y†ˆf1 exp‰ g y†Šg n
F Y n
exp‰g u n †Š exp‰ g y†Š
n
ˆ 1
n 7:98†
n

expf ‰g y† g u n †Šg
ˆ 1 :
n
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

g y†ˆ g u n † ‡ n y u n † ‡ ; 7:99†

where n ˆ dg(y)/dy is evaluated at y ˆ 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
exp‰ n y u n †Š
n
y†ˆ 1 ; 7:100†
F Y n
n
in which n and u n are functions only of n and not of y. Using the identity
c n

lim 1 ˆ exp c†;
n!1 n
for any real c, Equation (7.100) tends, as n !1
, to
F Y y†ˆ expf exp‰ y u†Šg; 7:101†

which was to be proved. In arriving at Equation (7.101), we have assumed that
as n !1 , F Y n (y) converges to F Y (y) as Y n converges to Y in some probabilistic
sense.

TLFeBOOK