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Q2. If a random variable, X, has a geometric distribution with parameter p âˆˆ (0,1), we write X ~ Geometric(p) and the probability mass function of X is p(1 - p)^(z-1) for z âˆˆ {1,2,3,...}. P(X = 2) = p(1 - p)^(2-1) = p(1 - p) if z = 2, and P(X = 2) = 0 otherwise. The expected value of X, denoted as E(X), is unknown and needs to be determined. Suppose that X1, X2, ..., Xn are independent and identically distributed random variables, where Xi ~ Geometric(p) for i âˆˆ {1,2,...,n}. The following estimator, D, is proposed as an estimator for p: if Xi = 1, then p = 1; otherwise, p = 0. Using the Rao-Blackwell theorem, we can find an unbiased estimator of p, denoted as p, such that Var(p) < Var(p). The Rao-Blackwell theorem states that if X1, X2, ..., Xn are independent and identically distributed Geometric(p) random variables, then Xi ~ NegBin(n, p) for i âˆˆ {1,2,...,n}.

          Q2. If a random variable, X, has a geometric distribution with parameter p âˆˆ (0,1), we write X ~ Geometric(p) and the probability mass function of X is p(1 - p)^(z-1) for z âˆˆ {1,2,3,...}. P(X = 2) = p(1 - p)^(2-1) = p(1 - p) if z = 2, and P(X = 2) = 0 otherwise. The expected value of X, denoted as E(X), is unknown and needs to be determined.

Suppose that X1, X2, ..., Xn are independent and identically distributed random variables, where Xi ~ Geometric(p) for i âˆˆ {1,2,...,n}. The following estimator, D, is proposed as an estimator for p: if Xi = 1, then p = 1; otherwise, p = 0. 

Using the Rao-Blackwell theorem, we can find an unbiased estimator of p, denoted as p*, such that Var(p*) < Var(p). The Rao-Blackwell theorem states that if X1, X2, ..., Xn are independent and identically distributed Geometric(p) random variables, then Xi ~ NegBin(n, p) for i âˆˆ {1,2,...,n}.

q2 if a random variable x has a geometric distribution with parameter p 01 we write x geometricp and the probability mass function of x is p1 pz 1 i 123 px 2 0 otherwise and ex suppose th 66656

Added by Michael W.

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