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Probability - Conditional Distribution

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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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This solution includes a detailed explanation of the law of total variance formula and conditional expectations by using this problem as an illustrative example. The step-by-step explanation in this solution provides a clear application of conditional expectation and variance formulas in a concrete setting.

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(a) Since the ...

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