Page 276 - Elements of Distribution Theory

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052184472Xc09 CUNY148/Severini June 2, 2005 12:8

262 Approximation of Integrals

Since
1
∞ ∞
x−1 j j
(1 − u) (yu) /j! ≤ y /j! = exp(y),
j=0 0 j=0
we may interchange summation and integration. Hence, by Theorem 9.2,

∞ j 1
y x−1 j
x
γ (x, y) = y exp(−y) (1 − u) u du
j!
j=0 0
∞ y ( j + 1) (x)
j
x

= y exp(−y) ,
j! (x + j + 1)
j=0
which, after simplification, is identical to (9.2).
Standard normal distribution function
Let
1 1 2
φ(z) = √ exp − z , −∞ < z < ∞
(2π) 2
denote the standard normal density function and let
z
(z) = φ(t) dt, −∞ < z < ∞
−∞
denote the standard normal distribution function.
Although (z)is defined by an integral over the region (−∞, z), calculation of (z),
for any value of z, only requires integration over a bounded region. Define

z
0 (z) = φ(t) dt, z ≥ 0.
0
The following result shows that (z) can be written in terms of 0 (z); also, by a change-
of-variable in the integral defining 0 it may be shown that 0 is a special case of the
incomplete gamma function γ (·, ·). The proof is left as an exercise.

Theorem 9.8.
 1
 − 0 (−z) if z < 0
2
(z) =
1
 + 0 (z) if z ≥ 0
2
and, for z > 0,
1 1 z 2
0 (z) = √ γ , .
2 π 2 2

Hence, using Theorem 9.8, together with the series expansions for γ (·, ·)given in The-
orem 9.7, we obtain the following series expansions for 0 (·); these, in turn, may be used
to obtain a series expansion for (·). The result is given in the following theorem, whose
proof is left as an exercise.