Answer:
Step-by-step explanation:
A negative binomial random variable "is the number X of repeated trials to produce r successes in a negative binomial experiment. The probability distribution of a negative binomial random variable is called a negative binomial distribution, this distribution is known as the Pascal distribution".
And the probability mass function is given by:
Where r represent the number successes after the k failures and p is the probability of a success on any given trial.
Solution to the problem
For this case the likehoof function is given by:
If we replace the mass function we got:
When we take the derivate of the likehood function we got:
And in order to estimate the likehood estimator for p we need to take the derivate from the last expression and we got:
And we can separete the sum and we got:
Now we need to find the critical point setting equal to zero this derivate and we got:
For the left and right part of the expression we just have this using the properties for a sum and taking in count that p is a fixed value:
Now we need to solve the value of from the last equation like this:
And if we solve for we got:
And if we divide numerator and denominator by n we got:
Since