Problem S5.5. Consider the following chemical reaction, A+MX+M and 2X+E+D, where molecules A, M, E, and D are obtained from large reservoirs and can be assumed constant.

(a) Find the probability to have n X molecules in the system after a very long time, too.

(b) Find the average number of X molecules after a very long time. [Some hints: Use the boundary conditions F(1) = 1 and F(-1) = 0 for the generating function, F(z). The generating function can be found in terms of modified Bessel functions. The transformation F(2) = SG(s), where s = (1+2)/2, might be helpful.]