Prob & Stats 20B: Maximum Likelihood Estimation for Geometric Distribution

Описание: What does Maximum Likelihood Estimation (M.L.E.) give for the parameter p of a geometric distribution? For a geometric random variable X, counting the number of trials until the first success, and probability of success p on each independent and identical trial, the probability mass function (PMF) is f(x) = p*(1-p)^(x-1). The likelihood function takes a random sample x1, x2, ..., xn and multiplies these PMF values to get L(p) = f(x1)*f(x2)*...*f(xn). It is easier to maximize the log-likelihood function LL(p) = ln(L(p)) because the products get converted to summations. In the end, we get the estimator p̂ = 1/X̄ after taking the derivative, setting it equal to zero, and solving for the critical point for p.

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