Problem 2 Under the same setting as Problem 1, suppose that D,- is assigned to the n units in a Bernoulli trial, that is each unit receive-2. treatment independently with probability Pr(D,- = 1) = p. Part a (lOpts) Recall that the number of treated units, N; is dened as N, = :1: l 1),. Recall that in the case of a Bernoulli trial, N; is a random variable. 1. What distribution does D, follow? 2. What is E[Nt]? 3. What is Vor[Nt]? 4 . Suppose that we wanted the expected number of treated units in our Bernoulli trial to be the same as the number of treated units as a completely randomized experiment with m treated units. What value of 30 should we choose? Part b (l5pts) After conducting the experiment as described above, we wish to estimate the ATE. To do so, we employ the following estimator: n A 1 D, 1D,- TIPW-gZ(KY ) 1. ,=, p 110 Show that under consistency, positivity, and ignorability for all treatments this estimator is unbiased for the ATE. Part c (5pts) Suppose now that we used the same estimator dened above in a completely randomized experiment, where exactly at units are treated. Show that, in this case, the estimator above is equal to the Neyman \"differenceinmeans\" estimator we saw in class