Maximum likelihood, also called the maximum likelihood method, is the procedure of finding the value of one or more parameters for a given statistic which makes the known likelihood distribution a maximum. The maximum likelihood estimate for a parameter  is denoted .

For a Bernoulli distribution,

(1)

so maximum likelihood occurs for . If  is not known ahead of time, the likelihood function is

			(2)
			(3)
			(4)

where  or 1, and , ..., .

(5)

(6)

Rearranging gives

(7)

so

(8)

For a normal distribution,

			(9)
			(10)

so

(11)

and

(12)

giving

(13)

Similarly,

(14)

gives

(15)

Note that in this case, the maximum likelihood standard deviation is the sample standard deviation, which is a biased estimator for the population standard deviation.

For a weighted normal distribution,

(16)

(17)

(18)

gives

(19)

The variance of the mean is then

(20)

But

(21)

so

			(22)
			(23)
			(24)

For a Poisson distribution,

(25)

(26)

(27)

(28)

REFERENCES:

Harris, J. W. and Stocker, H. "Maximum Likelihood Method." §21.10.4 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, p. 824, 1998.

Hoel, P. G. Introduction to Mathematical Statistics, 3rd ed. New York: Wiley, p. 57, 1962.

Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Least Squares as a Maximum Likelihood Estimator." §15.1 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 651-655, 1992.