Page 277 - Elements of Distribution Theory

P1: JZP
052184472Xc09 CUNY148/Severini June 2, 2005 12:8

9.2 Some Useful Functions 263

Theorem 9.9. For z ≥ 0,
∞
∞
1 (−1) j z 2 j+1 1 −z 2 z 2 j+1
0 (z) = √ = √ exp .
(2π) 2 j! 2 j + 1 2 2 2 2 ( j + 3/2)
j
j
j=0 j=0
Since φ(z) > 0 for all z, (·)isa strictly increasing function and, since it is a distribution
function,
lim (z) = 1 and lim (z) = 0.
z→∞ z→−∞
The following result gives some information regarding the rate at which (z) approaches
these limiting values; the proof follows from L’Hospital’s rule and is left as an exercise.

Theorem 9.10.
1 − (z) (z)
lim = 1 and lim =−1.
z→∞ φ(z)/z z→−∞ φ(z)/z

The following result gives precise bounds on
1 − (z) (z)
and
φ(z)/z φ(z)/z
that are sometimes useful.

Theorem 9.11. For z > 0,
z 1
φ(z) ≤ 1 − (z) ≤ φ(z).
1 + z 2 z
For z < 0,

|z| 1
φ(z) < (z) < φ(z).
1 + z 2 |z|
Proof. Fix z > 0. Note that

∞ ∞ 1 1 ∞
2 2 2
exp(−x /2) dx = x exp(−x /2) dx ≤ x exp(−x /2) dx.
z z x z z
Since
d 2 2
exp(−x /2) =−x exp(−x /2),
dx

∞
2
2
x exp(−x /2) dx = exp(−z /2)
z
so that

∞ 1
2 2
exp(−x /2) dx =≤ exp(−z /2)
z z
or, equivalently,
1
1 − (z) ≤ φ(z).
z