Bacteria in suspension form into clumps, the probability that a random clump will contain n bacteria is 0-1(1-0) (n = 1, 2, ...). When subject to harmful radiation there is a probability At that a bacteria will be killed in any interval of length 8t irrespective of the age of the bacteria in the same or different clumps. A clump is not killed until all the bacteria in it have been killed. Prove that the probability that a random clump will be alive after being exposed to radiation for a time t is e-(1-0+0e). In order to estimate the strength, A, of the radiation a unit volume of the unradiated suspension is allowed to grow and the number n of live clumps counted. The remainder of the suspension is irradiated for a time and then a unit volume is allowed to grow free of radiation and the number r of live clumps counted. It may be assumed that both before and after radiation the clumps are distributed randomly throughout the suspension. The experiment is repeated with new suspensions s times in all giving counts (n, na, na) and (r, ra,, ). Show that if 0 is known the maximum likelihood estimate of A is where 1-In N-OR) \R(1-0))" N=n, and R=r 1-1 (Camb. Dip.) i-1