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Recall, the p.m.f. of the geometric(p) distribution is p(x) = p(1 -p)*, x=0, 1, 2,..., with E(X) = (1 - p)/p and V(X) = (1 -p)/p2. Xis the number of failures before the first success in a sequence of independent Bernoulli trials, each with probability of success p. Suppose we have n draws from a geometric(p) distribution, x1,....Xn, with p unknown. a. Show that the maximum likelihood estimate of p is equal to P = b. Researchers have 100 random rats run a maze until they can each exit the maze in no more than 2 minutes. The number of failed attempts before each rat's first successful attempt is recorded. Number Failures Before First Success 0 1 2 3 Observed Frequency 52 29 15 4 Is a geometric distribution plausible? Perform a chi-squared goodness of fit test for this data. Use a = 0.05 and group {x 2 3} into a single cell