Consider the probability function of the (????) distribution

i) Show that f (x) can be calculated via the recursion

where a = 0 and b = ????, with initial condition f (0) = e−????.
(ii) Verify that f (x) > f (x − 1) if and only if x (iii) Prove that

(a) if ???? is not an integer, then f attains its maximum value at the point x0 = [????];

(b) if ???? is a positive integer, then f has two maxima, attained at the points x0 = ???? and x′0 = ???? − 1.

f(x) = P(X = x)=e x! = 0,1,2,...