Marginal distribution

As marginal distributions or marginal distribution, the probability distributions of subfamilies of a given family of random variables are referred to in the stochastics. The distribution of the entire family is called to clarify and joint distribution of the random variables. If, for example, and random variables ( on the same probability space ), then the names of the distributions of the individual variables and the marginal distributions of the random vector.

Marginal distributions can be calculated for both discrete as well as continuous variables. As with distributions generally a distinction is accordingly:

  • Discrete marginal distributions
  • Continuous marginal distributions

In addition, one can form the marginal distribution for both absolute frequencies and relative frequencies for. The individual values ​​of the marginal distribution is then called marginal frequencies (also marginal frequencies or marginal frequencies ). The marginal frequencies for categorical divided ( distinct ) features can be seen at the edge of a contingency table. You are here the sums of the frequencies on the neglected feature of time.

Example based on contingency tables

Marginal distributions of discrete features can be represented in contingency tables. At the edge of this table can be the marginal frequencies, which together form the marginal distribution, read as sums over the neglected feature.

For example, here is to see with absolute frequencies a contingency table. The same could also be achieved with relative frequencies.

To be the edge frequency in class 10 under the neglect of whether one is male or female is 20, the corresponding edge frequency for class 11 is also 20 The marginal distribution is thus uniformly distributed because there are equal numbers of students in both classes. The feature class is distinct, that is, in well-defined categories divided.

However, there are features that are not divided into categories, such as body size. These features are continuous, because there are smooth transitions between all possible values ​​of the feature. Such features can not be represented in tables. In order to enable the representation of a contingency table but it is possible in the feature classes (meaning here categories) divide, by defining so-called class boundaries. The steady feature of body size could be divided by specifying a class boundary 142cm and the people in people > 142cm and 142cm ≤ divides. For this banded class groups can now be used again to measure the class frequencies in a contingency table enters. As no person in a class ( > 142) is also the same in another class ( ≤ 142) can be one also speaks of a division into disjoint sets.

Definition

For random variables with common distribution function is called for each of the function and a (one-dimensional ) marginal distribution of.

According to one -dimensional marginal distributions can be defined by only the infinite sets.

Which belongs to the marginal distribution density distribution is referred to as edge density or marginal density. A marginal density from the joint density is obtained by summation or integration over the no longer considered variables.

Properties

  • A system of jointly distributed random variables has -dimensional marginal distributions.
  • For stochastically independent random variables, the joint distribution is the product of the marginal distributions.
  • The marginal distributions of a Gaussian distribution function are also Gauß'sch.

Discrete Distributions

Are and discrete random variables on the same probability space, their joint distribution by

Is given, then the marginal distribution of calculated by summing over all possible values ​​of

Analogously, for the marginal distribution of

Continuous distributions

Are and continuous random variables on the same probability space with joint density function, we obtain the marginal densities of the and by integrating over the other variables:

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