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