A pollster wishes to obtain information on intended voting behavior in a two party system and samples a fixed number (n) of voters. Let X_1 ..., X_n denote the sequence of independent Bernoulli random variables representing voting intention, where E(X_l) = p, i = 1, ..., n
(a) Suppose the number of voters n is fixed, compute E(X) and Var(X)
(b) Suppose the number of voters N is a random variable, with P(N=n)=p_n
(i) Find the conditional distribution of X given N
(ii) Find the conditional expectation E(X|N) and conditional variance Var(X|N)
(c) Express Var(X) in terms of p, E(N), and Var(N), by using the law of total variance.
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