Bimodal Distribution - Mixture Distributions

Mixture Distributions

A bimodal distribution most commonly arises as a mixture of two different unimodal distributions (i.e. distributions having only one mode). In other words, the bimodally distributed random variable X is defined as with probability or with probability where Y and Z are unimodal random variables and is a mixture coefficient. For example, the bimodal distribution of sizes of weaver ant workers shown in Figure 2 arises due to existence of two distinct classes of workers, namely major workers and minor workers. In this case, Y would be the size of a random major worker, Z the size of a random minor worker, and α the proportion of worker weaver ants that are major workers.

A mixture of two normal distributions has five parameters to estimate: the two means, the two variances and the mixing parameter. A mixture of two normal distributions with equal standard deviations is bimodal only if their means differ by at least twice the common standard deviation. Estimates of the parameters is simplified if the variances can be assumed to be equal (the homoscedastic case).

Mixtures of other distributions require additional parameters to be estimated.

A mixture of two unimodal distributions with differing means is not necessarily bimodal. The combined distribution of heights of men and women is sometimes used as an example of a bimodal distribution, but in fact the difference in mean heights of men and women is too small relative to their standard deviations to produce bimodality.

Bimodal distributions have the peculiar property that - unlike the unimodal distributions - the mean may be a more robust sample estimator than the median. This is clearly the case when the distribution is U shaped like the arcsine distribution. It may not be true when the distribution has one or more long tails.