32. Let the variables X; (i = 1, . . . , s) be independently distributed with Poisson distribution P( i)' For testing the hypothesis H :'L~j .s a (for example, that the combined radioactivity of a number of pieces of radioactive material does not exceed a), there exists a UMP test, which rejects when 'LX; > C. [If the joint distribution of the X's is factored into the marginal distribution of 'LX; (Poisson with mean 'L~j) times the conditional distribution of the variables Y; = X;/'LX; given 'LX; (multinomial with probabilities Pi = ~;/'L~j)' the argument is analogous to that given in Example 8.]