**EXPONENTIAL DISTRIBUTION**

Let

The likelihood function is given by

The natural logarithm of

Thus,

Note that,

Hence,

**GEOMETRIC DISTRIBUTION**

Let

The likelihood function is given by

The natural logarithm of

Thus restricting *p* to

Solving for *p*, we obtain *p* is

Again this estimator is the method of moments estimator, and it agrees with the intuition because, in n observations of a geometric random variable, there are *n* successes in the

**NORMAL DISTRIBUTION**

Let

The partial derivatives with respect to

The equation

By considering the usual condition on the second partial derivatives, these solutions do provide a maximum. Thus the maximum likelihood estimators

Where we compare the above example with the introductory one, we see that the method of moments estimators and the maximum likelihood estimators for

**BINOMIAL DISTRIBUTION**

Observations: *k* successes in *n* Bernoulli trials.

**POISSON DISTRIBUTION**

Observations: