
245Stochastic or Random Processes and Time Series
Another pdf typically employed to model rainfall amount is the Gamma pdf
px
xb xb
bc
c
()
()exp( )
()
=
−
−
1
Γ
(7.20)
where the parameters c and b are shape and scale, respectively. The scale is the inverse of the rate
a = 1/b. The mean and variance are
cb cb
(7.21)
For the gamma, it is relatively easy to generate rainfall with a given mean and variance of rain since
we can solve for c and b from Equation 7.21
σ
µ
µ
σ
X
X
X
X
b
b
and
and
2
2
2
(7.22)
There is no closed form equation for the cdf unless the shape parameter c is an integer and in this
case Γ(c) = (c − 1)! and the pdf reduces to the ...