Page 310 - Elements of Distribution Theory

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296 Approximation of Integrals

9.3 Prove Theorem 9.5.

Exercises 9.4 and 9.5 use the following definition.
The incomplete beta function is defined as
x

I(r, s, x) = t r−1 (1 − t) s−1 dt,
0
where r > 0, s > 0, and 0 ≤ x ≤ 1.
9.4 Show that, for all r > 0, s > 0, and 0 ≤ x ≤ 1,
1 − I(r, s, x) = I(s,r, 1 − x).
9.5 Show that, for all r > 1, s > 1, and 0 ≤ x ≤ 1,
I(r, s, x) = I(r, s − 1, x) − I(r + 1, s − 1, x).
9.6 Prove Theorem 9.8.
9.7 Prove Theorem 9.9.
9.8 Prove Theorem 9.10.
9.9 Suppose that, as n →∞,

a 1 a 2 1
f (n) = a 0 + + + O
n n 2 n 3
and

b 1 b 2 1
g(n) = b 0 + + + O
n n 2 n 3
for some constants a 0 , a 1 , a 2 , b 0 , b 1 , b 2 such that b 0 = 0.
Find constants c 0 , c 1 , c 2 such that
f (n) c 1 c 2 1
= c 0 + + + O .
g(n) n n 2 n 2
9.10 Show that
x
(x + 1) = lim z β(x, z), x > 0.
z→∞
9.11 Let X denote a random variable with an absolutely continuous distribution with density function
β α
x α−1 exp(−βx), x > 0.
(α)
Let h denote a function such that
∞

( j)
h(t) = h (0)t j
j=0
2
and such that h(t) = O(exp(at )) as |t|→∞ for some constant a. Find an asymptotic expansion
for E[h(X)] as β →∞, with α remaining fixed.
9.12 Let (·, ·) denote the incomplete gamma function. Show that, for fixed x,

x − 1 (x − 1)(x − 2) 1
(x, y) = y x−1 exp(−y) 1 + + + O
y y 2 y 3
as y →∞.
9.13 Let Y denote a real-valued random variable with an absolutely continuous distribution with
density function
p(y; a) = c(a)exp{−2a cosh(y)}, −∞ < y < ∞;

see Example 9.10. Find an approximation to E[cosh(Y)] that is valid for large a.