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Expected value

Lecture



The mathematical expectation is the average value of a random variable (this is the probability distribution of a random variable, considered in probability theory) [1] . In English literature it is denoted by   Expected value [2] (for example, from the English. Expected value or German Erwartungswert ), in Russian -   Expected value (perhaps from the English. Mean value or German. Mittelwert , and possibly from the "Mathematical expectation"). The statistics often use the designation   Expected value .

Content

  • 1 Definition
  • 2 Basic formulas for expectation
    • 2.1 Mathematical expectation of a discrete distribution
      • 2.1.1 Mathematical expectation of an integer value
    • 2.2 Mathematical expectation of absolutely continuous distribution
  • 3 Mathematical expectation of a random vector
  • 4 Mathematical expectation of converting a random variable
  • 5 The simplest properties of mathematical expectation
  • 6 Additional properties of mathematical expectation
  • 7 Examples
  • 8 Notes
  • 9 See also
  • 10 Literature

Definition [edit]

Let probabilistic space be given   Expected value and a random variable defined on it   Expected value . That is, by definition,   Expected value - measurable function. If there is a Lebesgue integral from   Expected value in space   Expected value , it is called the expectation, or average (expected) value, and is denoted by   Expected value or   Expected value .

  Expected value

Basic formulas for expectation [edit]

  • If a   Expected value - the distribution function of a random variable, then its expectation is given by the Lebesgue – Stieltjes integral:
  Expected value .

Mathematical expectation of a discrete distribution [edit]

  • If a   Expected value - discrete random variable with distribution
  Expected value ,

then it follows directly from the definition of the Lebesgue integral that

  Expected value .

Mathematical expectation of an integer value [edit]

  • If a   Expected value - positive integer random variable (special case discrete), having a probability distribution
  Expected value

then its expectation can be expressed in terms of the generating function of the sequence   Expected value

  Expected value

as the value of the first derivative in the unit:   Expected value . If the expectation   Expected value endlessly then   Expected value and we will write   Expected value

Now take the generating function   Expected value sequences of tails of distribution   Expected value

  Expected value

This generating function is associated with a previously defined function.   Expected value property:   Expected value at   Expected value . From this, by the mean-theorem, it follows that the expectation is equal to just the value of this function in the unit:

  Expected value

Mathematical expectation of absolutely continuous distribution [edit]

  • The expectation of an absolutely continuous random variable whose distribution is given by the density   Expected value equal to
  Expected value .

Mathematical expectation of a random vector [edit]

Let be   Expected value - random vector. Then by definition

  Expected value ,

that is, the expectation of a vector is determined componentwise.

Mathematical expectation of the conversion of a random variable [edit]

Let be   Expected value - Borel function, such that a random variable   Expected value has a finite mathematical expectation. Then the formula is valid for him:

  Expected value ,

if a   Expected value has a discrete distribution;

  Expected value ,

if a   Expected value has an absolutely continuous distribution.

If the distribution   Expected value random variable   Expected value general view then

  Expected value .

In the special case when   Expected value , Expected value   Expected value called   Expected value th moment of a random variable.

The simplest properties of the expectation [edit]

  • The mathematical expectation of a number is the number itself.
  Expected value
  Expected value - constant;
  • The expectation is linear, that is
  Expected value ,
Where   Expected value - random variables with finite mathematical expectation, and   Expected value - arbitrary constants;
  • Expectation preserves inequalities, that is, if   Expected value almost sure and   Expected value - a random variable with a finite mathematical expectation, then a mathematical expectation of a random variable   Expected value also of course
  Expected value ;
  • The expectation does not depend on the behavior of the random variable at the probability zero event, that is, if   Expected value almost surely
  Expected value .
  • Mathematical expectation of the product of two independent random variables   Expected value equal to the product of their mathematical expectations
  Expected value .

Additional properties of expectation [edit]

  • Markov's inequality;
  • Levi's theorem on monotone convergence;
  • Lebesgue's theorem on majorized convergence;
  • Wald's identity;
  • Lemma Fatou.
  • Mathematical expectation of a random variable   Expected value can be expressed through its moments generating function   Expected value as the value of the first derivative at zero:   Expected value

Examples [edit]

  • Let the random variable have a discrete uniform distribution, that is,   Expected value Then her expectation is
  Expected value

equal to the arithmetic average of all received values.

  • Let the random variable has a continuous uniform distribution on the interval   Expected value where   Expected value . Then its density is   Expected value and the expectation is
  Expected value .
  • Let the random variable   Expected value has a standard Cauchy distribution. Then
  Expected value ,

i.e. expectation   Expected value undefined.

Notes [edit]

  1. “Mathematical Encyclopedia” / Editor-in-chief I. M. Vinogradov. - M .: "Soviet Encyclopedia", 1979. - 1104 p. - (51 [03] M34). - 148 800 copies.
  2. A. N. Shiryaev 1 // “Probability”. - M .: MTSNMO, 2007. - 968 p. - ISBN 978-5-94057-036-3, 978-5-94057-106-3, 978-5-94057-105-6.

See also [edit]


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Probability theory. Mathematical Statistics and Stochastic Analysis

Terms: Probability theory. Mathematical Statistics and Stochastic Analysis