Variance

In the stochastic variance of a random variable is a measure of dispersion of, that is a measure of the deviation of a random variable from its expected value. The variance of the random variable is normally recorded as, or simply; it is always ≥ 0

The variance is a property of the distribution of a random variable and does not depend on chance. It measures the dispersion of the values ​​relative to the expected value, while the squares of the deviations are weighted according to their probability. In practice, the variance of the random variable with a variance estimator, such as the its empirical counterpart, the sampling variance is estimated.

• 2.1 shift kit
• 2.2 Linear Transformation
• 2.3 Variance of sums of random variables
• 2.4 Characteristic Function
• 2.5 moment generating function

• 3.1 Discrete random variable
• 3.2 Continuous random variable

Definition

It is a real random variable, which is integrable, ie it applies. Then there is their expectation, and we define the variance of as follows:

Is square integrable, ie, the variance is finite.

The variance is thus the second central moment of a random variable and thus the expected squared deviation of the random variable from its expected value.

The square root of the variance is called standard deviation:

Calculation for discrete random variables

A real random variable with a finite or countably infinite range of values ​​is called discrete. Your variance is then calculated as

Here, the probability that the assumed value, and

The expectation value. The buzz each extending over all the values ​​that can assume the random variable.

Calculation for continuous random variables

If a random variable has a probability density function that applies

In which

Calculation rules

Shift theorem

Variances can often be easier with the help of the shifting theorem

Calculate, since it must be determined for this purpose than the expected value of only the expected value of.

On the computer, this type of calculation, however, be avoided, as it can easily lead to catastrophic cancellation when using floating point numbers.

Linear transformation

For real numbers and is

This can be derived by means of the displacement law:

In particular, for the following and

Variance of sums of random variables

For the variance of an arbitrary sum of random variables is generally

Referred to herein is the covariance of the random variables, and it was used and that the following applies. Especially for two random variables and results, for example

Are the random variables pairwise uncorrelated, ie their covariances are all equal to zero, then it follows:

This set is also called the equation of Bienaymé. He is especially true if the random variables are independent, because independence follows from uncorrelated.

Characteristic function

The variance of a random variable can be represented with the help of its characteristic function. Ways and it follows with the shift theorem

Moment generating function

As for the moment generating function of the context

Applies, can the variance in order to calculate the following way:

Examples

Discrete random variable

Given a discrete random variable which the values ​​, and each with the probabilities, and accepts. Is the expected value

And the variance is therefore

With the shift theorem is also obtained

For the standard deviation, this resulted in

Continuous random variable

A continuous random variable has the density function

With the expected value

To calculate the variance using the shifting theorem as

Generalizations

In case of a real random vector with square integrable components, the variance generalized to the covariance matrix of a random vector:

Here, the vector of the expected values ​​. The entry of the th row and th column of the covariance matrix. In the diagonal so are the variances of the individual components.

Analogous to conditional expected values ​​can be in the presence of additional information, such as the values ​​of other random variables, conditional variances look.