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Probability distribution

Lecture



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

Content

  • 1 Definition
  • 2 Ways to assign distributions
    • 2.1 Discrete Distributions
    • 2.2 Continuous Distributions
    • 2.3 Absolutely continuous distributions
  • 3 Notes

Definition [edit]

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

  Probability distribution

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

Ways to assign distributions [edit]

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

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

  1.   Probability distribution - non-decreasing function;
  2.   Probability distribution ;
  3.   Probability distribution 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   Probability distribution implies

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

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

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

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

Theorem 3. Discrete distribution has the following properties:

one.   Probability distribution ;

2   Probability distribution .

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

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

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

Theorem 4. If the function   Probability distribution such that:

  1.   Probability distribution ;
  2.   Probability distribution ,

then there is a distribution   Probability distribution such that   Probability distribution 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   Probability distribution - continuous distribution density, and   Probability distribution - its cumulative function, then

  1.   Probability distribution
  2.   Probability distribution .
created: 2015-01-17
updated: 2021-01-10
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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis