Let S X ++X be the sum of mutually independent variables each assuming the values 1, 2,..., a with probability 1/a. Show that the generating function is given by whence for jn P(s) = 00 (s(1-sa)" a(1 s) - P{Sja(-1)+-n-a" ( 1)*+3- y=0 00 -n ()() - - n-av = n-1 (Only finitely many terms in the sum are different from zero.)

Note: For a = 6 we get the probability of scoring the sum j + n in a throw with n dice. The solution goes back to De Moivre.