Let x_(1), x_(2), ..., x_(n) denote n independent and identically distributed Bernoulli random variables such that P(x_(i) = 1) = p and P(x_(i) = 0) = 1 - p, for each i = 1, 2, ..., n. Show that ∑_(l=1)^n x_(l) is sufficient for p by using the factorization criterion given below, factored into two nonnegative functions, L(y_(1), y_(2), ..., y_(n)|θ) = g(u, θ) × h(y_(1), y_(2), ..., y_(n)).
Let X_(1), ..., X_(n) denote n independent and identically distributed Bernoulli random variables such that P(X = 1) = p and P(X = 0) = 1 - p, for each i = 1, 2, ..., n. Show that X is sufficient for p by using the factorization criterion given below, factored into two nonnegative functions, L(y_(1), y_(2), ..., y_(n)) = g(u, x) × h(y_(1), y_(2), ..., y_(n)), where g(u) is a function only of u and x, and h(y_(1), ..., y_(n)) is not a function of x.
The likelihood function is L = p^x(1-p)^(1-x). By the factorization criterion above, X is sufficient for p with g(x) = p^x(1-p)^(1-x) and h(y) = 1.
The likelihood function is L = p^x(1-p)^x. By the factorization criterion above, x is sufficient for p with g(x) = p^x(1-p)^x and h(y) = p^x(1-p)^x.
