
16 Models of distribution functions F(x) and force of mortality µ(x)
where λ is the scale parameter and
Γ(k) =
Z
∞
0
x
k−1
e
−x
dx
is the gamma function of “shape parameter” k. The gamma distribution
function is the integral
F(x) =
λ
k
Γ(k)
Z
x
0
y
k−1
e
−λy
dy, x ≥ 0. (2.9)
For the gamma function see, e.g., Abramowitz and Stegun [1964]; for
the gamma distribution function see, e.g., Johnson et al. [1994] and
Johnson et al. [1995]. When k = 1, F becomes an exponential distribu-
tion function while for k = n/2 with integer n and λ = 1/2 it becomes
a chi-square distribution function with n degrees of freedom. We use
both of these distributions in this book.
For integer values ...