Probability distribution

The probability distribution is a law that describes the range of values ​​of a random variable and the probability of their outcome (appearance).

Definition [edit]

Definition 1. Let probability space be given and a random variable is defined on it . In particular, by definition, is a measurable mapping of measurable space in a measurable space where denotes a Borel sigma-algebra on . Then a random variable induces a probability measure on in the following way:

Measure called a random variable distribution . In other words, , in this way sets the probability that a random variable falls into the set .

Ways to assign distributions [edit]

Definition 2. Function is called the (cumulative) distribution function of a random variable . From the properties of probability follows

Theorem 1. The distribution function any random variable satisfies the following three properties:

1. - non-decreasing function;
2. ;
3. continuous on the right.

From the fact that a Borel sigma-algebra on a real line is generated by a family of intervals of the form implies

Theorem 2. Any function satisfying the three properties listed above is a distribution function for some distribution .

For probability distributions with certain properties, there are more convenient ways to set it.

Discrete distributions [edit]

Definition 3. A random variable is called simple or discrete if it takes no more than a countable number of values. I.e where - splitting .

The distribution of a simple random variable is then defined by definition: . Enter the designation can set function . It's obvious that . Using countable additivity it is easy to show that this function uniquely determines the distribution .

Definition 4. Function where often referred to as a discrete distribution .

Example 1. Let function set in such a way that and . This function sets the distribution of the random variable. , for which (Bernoulli distribution).

Theorem 3. Discrete distribution has the following properties:

one. ;

2 .

Continuous Distributions [edit]

A continuous distribution is a distribution that does not have atoms.

Absolutely continuous distributions [edit]

Absolutely continuous are the distributions having a probability density. The cumulative function of such distributions is absolutely continuous in the sense of Lebesgue.

Definition 5. The distribution of a random variable. called absolutely continuous if there is a non-negative function such that . Function then called the distribution density of a random variable .

Example 2. Let when and - otherwise. Then , if a .

Obviously, for any density distribution true equality . True and reverse

Theorem 4. If the function such that:

1. ;
2. ,

then there is a distribution such that is its density.

Simply applying the Newton-Leibniz formula leads to a simple relationship between the cumulative function and the density of an absolutely continuous distribution.

Theorem 5. If - continuous distribution density, and - its cumulative function, then

1. 
2. .