In Prob. 8, show that f(1) + f(2) + · · · = 1 (as it should be!). Calculate independently of Prob. 8 the maximum likelihood of p in Prob. 8 corresponding to a single observed value of X.

Data from Prob. 8

Let X = Number of independent trials until an event A occurs. Show that X has a geometric distribution, defined by the probability function f(x) = pq^x-1, x = 1, 2, · · ·, where p is the probability of A in a single trial and q = 1 - p. Find the maximum likelihood estimate of p corresponding to a sample x₁, x₂, · · ·, x_n of observed values of X.