Suppose that there are n trials x1, x2,...,xn from a Bernoulli process with parameter p, the probability of a success. That is, the probability of r successes is given by n r pr(1 − p) n−r. Work out the maximum likelihood estimator for the parameter p.
Question
Suppose that there are n trials x1, x2,...,xn from a Bernoulli process with parameter p, the probability of a success. That is, the probability of r successes is given by n r pr(1 − p) n−r. Work out the maximum likelihood estimator for the parameter p.
Solution
The maximum likelihood estimator (MLE) for the parameter p in a Bernoulli process can be derived as follows:
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The likelihood function for a Bernoulli process is given by L(p; x) = p^r * (1 - p)^(n - r), where r is the number of successes and n is the total number of trials.
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To find the MLE, we need to maximize this likelihood function. This is often done by taking the natural logarithm of the likelihood function, which simplifies the math without changing the value of p that maximizes the function.
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The log-likelihood function is therefore l(p; x) = log(L(p; x)) = r*log(p) + (n - r)*log(1 - p).
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To find the maximum of this function, we take the derivative with respect to p and set it equal to zero. This gives us the equation r/p - (n - r)/(1 - p) = 0.
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Solving this equation for p gives us the MLE for p, which is p = r/n.
So, the maximum likelihood estimator for the parameter p in a Bernoulli process is the ratio of the number of successes to the total number of trials.
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