Consider the observation X from the negative binomial distribution given in Section 5.4. Find the maximum likelihood estimator for p, assuming k is known.

The negative binomial distribution is a discrete probability distribution of the number of successes in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of failures (denoted k) occurs.

The probability mass function of the negative binomial distribution is given by:

P(X = x) = C(x + k - 1, x) * (1 - p)^(k) * p^(x)

where:

• x is the number of successes,
• k is the number of failures,
• p is the probability of success on an individual trial,
• C(n, r) is the binomial coefficient.

The likelihood function L(p; X) for a sample X = (x1, x2, ..., xn) is given by the product of the individual probabilities:

L(p; X) = Π P(X = xi) = Π C(xi + k - 1, xi) * (1 - p)^(k) * p^(xi)

Taking the natural logarithm of the likelihood function gives the log-likelihood function:

l(p; X) = Σ log(C(xi + k - 1, xi)) + k * log(1 - p) + xi * log(p)

The maximum likelihood estimator (MLE) for p is found by taking the derivative of the log-likelihood function with respect to p, setting it equal to zero, and solving for p:

dl/dp = -k/(1 - p) + Σ xi/p = 0

Solving this equation for p gives the MLE for p:

p_hat = Σ xi / (n * k + Σ xi)

where p_hat is the MLE for p, n is the number of observations, and Σ xi is the sum of the observations.