The beta distribution (the term first used by Gini, 1911) is defined as:

Beta Distribution (animation)

The animation above shows the beta distribution as the two shape parameters change.

A distribution that has two modes (thus two "peaks").

Bimodality of the distribution in a sample is often a strong indication that the distribution of the variable in population is not normal. Bimodality of the distribution may provide important information about the nature of the investigated variable (i.e., the measured quality). For example, if the variable represents a reported preference or attitude, then bimodality may indicate a polarization of opinions. Often however, the bimodality may indicate that the sample is not homogenous and the observations come in fact from two or more "overlapping" distributions. Sometimes, bimodality of the distribution may indicate problems with the measurement instrument (e.g, "gage calibration problems" in natural sciences, or "response biases" in social sciences).

See also unimodal distribution , multimodal distribution .

The binomial distribution (the term first used by Yule, 1911) is defined as:

f(x) = [n!/(x!*(n-x)!)] * p^{x} * q^{n-x}

for x = 0, 1, 2, ..., n

where

p is the probability of success at each trial

q is equal to 1-p

n is the number of independent trials

Two variables follow the bivariate normal distribution if for each value of one variable, the corresponding values of another variable are normally distributed . The bivariate normal probability distribution function for a pair of continuous random variables (X and Y) is given by:

where

_{1},

_{2} are the respective means of the random variables X and Y

_{1},

_{2} is the correlation coefficient of X and Y

is the base of the natural logarithm, sometimes called Euler's e (2.71...)

is the constant Pi (3.14...)

See also, Normal Distribution , Elementary Concepts (Normal Distribution)