How Do You Calculate Maximum Likelihood Estimates for Different Distributions?

[probabilityst]

I actually have two questions, both of which are on the same topic

Homework Statement

Consider X = number of independent trials until an event A occurs. Show that X has the probability mass function f(x) = (1-p)^x p, x = 1,2,3,..., where p is the probability that A occurs in a single trial. Also show that the maximum likelihood estimate of p(^p) is 1/(sample mean), where (sample mean) = (sum from 1 to n of x)/n. This experiment is referred to as the negative binomial experiment

2. Find the maximum likelihood estimate for the paramter mu of the normal distribution with known variance sigma^2 = sigma_0_^2
_0_ is subscript

Homework Equations

The Attempt at a Solution

Um, I'm not actually sure how to go about solving this problem. Is maximum likelihood estimate similar to minimum variance unbiased estimator?

Any help is greatly appreciated, thank you

 

[HallsofIvy]

No. The "maximum likelihood estimator" for a parameter is the value of the parameter that gives the largest possible probability for the actual sample.