Mixture distribution

The term mixed distribution or composite distribution comes from the theory of probability.

It is considered to introduce a feature that derives from two populations and. In the stochastic model, so it is actually two random variables and, often from the same distribution type. The proportions of the two characteristics are with. The distribution is observed, which is determined by the total probability

Or as a mixed distribution with the corresponding distribution functions

Definition

Can the density function of a continuous random variable as

Write, it is said that a mixed distribution follows. Here are the density functions of continuous random variables and the probabilities with

One can easily show that under these conditions is non-negative and

Met.

Accordingly, the probability function of a discrete mixing distribution as follows

From the probability functions of discrete random variables.

Properties

For the moments of X holds:

This follows from ( in the continuous case )

A similar calculation gives the formula for the discrete case.

Example

A trout trout farmers sold in large quantities. It is made in the autumn when emptying the ponds an inventory. The trout are fished out weighed. This results in the distribution of the weight, as can be seen in the graph. The Zweigipfligkeit the distribution suggests a mixed distribution. It turns out that the trout come from two different ponds. The trout weights from the first pond is normally distributed with mean 400 g and the variance g2 4900 and those from the second pond with the expected value of 600 g and the variance g2 8100. From the first pond are 40% of the trout, from the second 60%. This gives the density function ( see figure).