states with two male senators, 15 states with one male and one female senator, and 4 states with two female senators. If we think of the voters in any given state as a slot machine choosing their senatorial delegation at random (with respect to gender), is the gender of the two senators dependent or independent? The log likelihoods of -47 and -48 are so close that it would appear as if there is no bias for or against same-gender senatorial pairs. (That the selection of senators is biased with respect to gender per se is self-evident.)5. (30 pts) Maximum Likelihood problems (ef. ASV $4.3) (a) (5 pts) Suppose X is distributed as a normal random variable with variance equal to one, and unknown mean . If you record N independent observations {X1, . .., Xx}, what is the best guess for the unknown mean , based on a maximum likelihood estimate? Show the details of your calculation. (b) (5 pts) Suppose Y is distributed as an exponential random variable with unknown pa- rameter A. If you record / independent observations {Y1, ..., Yw}, what is the best guess for the unknown parameter A, based on a maximum likelihood estimate? Show the details of your calculation. (c) (20 pts) Suppose a slot machine produces a pair of tokens drawn from the set { Apple, Banana We can write the outcome Z as either AA, AB, BA or BB. Assume the order does not matter, so AB and BA are equivalent outcomes. Suppose we observe 50 independent trials of this machine, and find that AA occurs 31 times, AB (or BA) occurs 15 times, and BB occurs 4 times. Use maximum likelihood to estimate the parameters of two contrasting models: Model I: Each token is chosen independently of the other with P(Token = 1) = p and P(Token = B) = 1 -p. Therefore P(Z = 14) = p', P(Z = AB or BA) = 2p(1 -p), and P(Z = BB) = (1 -p). Model II: The tokens may not be chosen independently. In this case P(Z = 14) = po, IP(Z = AB or BA) = p1, and P(Z = BB) = p2 where po + pit p2 = 1. For each model, formulate the likelihood function L for the observation that { Z1, . .., Zso} include 31 counts of AA, 15 counts of (AB or BA), and 4 counts of BB. For Model I, Ly should be a function of p. For Model II, Ly should be a function of po and pi (since p2 = 1 - po + pi is determined by the first two parameters). After finding p, the maximum likelihood estimate for model I, and (P1, p2) for model II, compare the log likelihoods log( Li(p)) versus log(Lu(po, pi)). Base on this data, does it seem plausible or implausible that the two tokens are chosen independently? Discuss. Hint: Under both models, the distribution of 50 independent samples among three alter- natives is governed by the multinomial distribution. Comparison and discussion. The inspiration for this problem is the following: As of October, 2018 the United States Congress had 100 senators, 24 of whom are women. There are 31