Let x have pmf p(x = k) = cpk k fork = 1, 2, . . where p is a parameter with 0 < p < 1 and c is a normalizing constant. we have c = −1/ log(1 − p), as seen from the taylor series − log(1 − p) = p + p2/2 + p3/3 + this distribution is called the logarithmic distribution (because of the log in the above taylor series), and has often been used in ecology.

(a) show that the pmf is a valid pmf.

(b) find the mean and variance of x.