Neither the gamma nor lognormal distribution can be mixed with another
gamma or lognormal distribution and the resuit still be gamma or lognormally distributed.
Formally, let f, (x; 62) and fo (x; 62) be either
gamma or lognormal probability density functions with parameter vectors
6, and 85, respectively.
Then assume we sample from a population with
probability p that the variate is distributed as f,(x; 9,).
The resulting variate is distributed as
pf, (x; 9,) + (1i-p) fox; 92)
The concept of mixing two distributions can be extended farther.
that the EX = u_
Suppose
of f(x) is actually drawn from a second density, atu, ).
Then the distribution of x is
_ £_(x,u_)
ox
x
fluOle) = GD
orf, (x,u,)
= Fly OH? gt)
f(x, hy, ) is a family of distributions indexed by the parameter uy (see
Mood et*al.,
1974:122-124).
This result can be applied to transuranic research by conceptualizing
the distribution of radioactivity in a fallout pattern.
Suppose we
stratify the fallout area into n strata, each with mean concentration
EX,, i=l, 2, ..., n.
If a random sample of 1-m* quadrats are taken from
strata i, the expected concentration will be EX,.
However, quadrats
closer to ground zero would be expected to have slightly larger concentrations on the average than quadrats farther away from ground zero.
However, this process is stochastic so one method of expressing this
randomness is to assume the expected value of a quadrat is actually
drawn from some distribution, g(u,)-
To simulate this process, a, )} was assumed to be a uniform distribution
with density function
Thus, the expected value of an observation was first drawn from a uniform
distribution, and then a variate generated from a gamma or lognormal
distribution with this expected value.
The expected value of the resulting distribution must be evaluated to show that indeed the expected
value is (a + b)/2:
614