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Expectation and variance

2022-06-25 01:19:00 【herbie】

Expectation and variance

1 background

The distribution function is the most complete description of the probability properties of random variables , The numerical characteristics of random variables are constants determined by the distribution of random variables , It characterizes random variables （ Or say , Its distribution is characterized ） The nature of a certain aspect of . When we understand the economic situation of workers in a certain industry , I am afraid the first concern will be its average income ( namely expect ), This gives us a general impression . Another important digital feature , Is to measure a random variable （ Or its distribution ） The degree of dispersion of values ( namely variance ).

2 Mathematical expectation

2.1 Definition

set up discrete A random variable The law of distribution of is ：

If the series

Absolute convergence ( namely ), It is called series The sum of is a discrete random variable Of Mathematical expectation , Write it down as , namely

set up Continuous type A random variable The probability density of is , If integral

Absolute convergence , It is called integral The value of is a continuous random variable Of Mathematical expectation , Write it down as , namely

2.2 nature

Suppose that the mathematical expectation of the random variable encountered exists , Then its expectation has the following important nature ：

nature 1： set up Is constant , Then there are

nature 2： set up It's a random variable , Is constant , Then there are

nature 3： set up It's two random variables , Then there are

This property can be extended to the case of the sum of any finite random variables .

nature 4： set up Are independent random variables , Then there are

This property can be extended to any finite product of independent random variables .

2.3 prove

prove 1： Set the random variable Constant , The probability density is , Then according to the expectation definition, we can get

Certificate completion .

prove 2： Set the random variable The probability density of is , Constant , Then according to the expectation definition, we can get

Certificate completion .

prove 3： Let two-dimensional random variables The probability density of is . The marginal probability density is , From the expectation of compound random variables

Certificate completion .

prove 4： And then prove 3, You Ruo and Are independent of each other ,

Certificate completion .

3 variance

3.1 Definition

set up It's a random variable , if There is , said by Of variance , Write it down as or , namely

The application also introduces a quantity , Write it down as , be called Standard deviation or Mean square error .

about discrete A random variable , Yes

among , yes The distribution law of .

about Continuous type A random variable , Yes

among , yes Probability density of .

3.2 nature

nature 1： set up Is constant , Then there are

nature 2： set up It's a random variable , Is constant , Then there are

nature 3： set up It's two random variables , Then there are

Specially , if Are independent of each other , Then there are

This property can be extended to the case of the sum of any finite number of independent random variables .

nature 4： If and only if With probability 1 Take the constant , namely

3.3 prove

prove 1：

prove 2：

prove 3：

The third item at the right end of the above formula ：

if Are independent of each other , from Mathematical expectation Of nature 4 It can be seen that the right end of the above formula is 0, therefore

prove 4： adequacy ： set up , Then there are , therefore

The need for ： set up , To prove . Using the method of disproportion , hypothesis , Then for a certain number , Yes , But by Chebyshev inequality （ See the previous article Proof and application of Chebyshev inequality ）, For arbitrary and , Available