variance
Definition
The variance of a real-valued random variable
X is
VarX=πΌ[(X-m)2],m=πΌX, |
provided that both expectations πΌX and πΌ[(X-m)2] exist.
The variance of X is often denoted by Ο2(X), Ο2X, or simply Ο2. The exponent on Ο is put there so that the number Ο=βΟ2 is measured in the same units as the random variable X itself.
The quantity Ο=βVarX is called the standard deviation
of X;
because of the compatibility of the measuring units,
standard deviation is usually the quantity that is quoted
to describe an emprical probability distribution, rather than the variance.
Usage
The variance is a measure of the dispersion or variation of a random variable about its mean m.
It is not always the best measure of dispersion for all random variables, but compared to other measures, such as the absolute mean deviation, πΌ[|X-m|], the variance is the most tractable analytically.
The variance is closely related to the π2 norm for random variables over a probability space.
Properties
-
1.
The variance of X is the second moment of X minus the square of the first moment:
VarX=πΌ[X2]-πΌ[X]2. This formula
is often used to calculate variance analytically.
-
2.
Variance is not a linear function. It scales quadratically, and is not affected by shifts in the mean of the distribution
:
Var[aX+b]=a2VarX, for any a,bββ. -
3.
A random variable X is constant almost surely if and only if VarX=0.
-
4.
The variance can also be characterized as the minimum of expected squared deviation of a random variable from any point:
VarX=inf -
5.
For any two random variables and whose variances exist, the variance of the linear combination
can be expressed in terms of their covariance
:
where , and .
-
6.
For a random variable , with actual observations , its variance is often estimated empirically with the sample variance
:
Title | variance |
Canonical name | Variance |
Date of creation | 2013-03-22 11:53:46 |
Last modified on | 2013-03-22 11:53:46 |
Owner | stevecheng (10074) |
Last modified by | stevecheng (10074) |
Numerical id | 14 |
Author | stevecheng (10074) |
Entry type | Definition |
Classification | msc 62-00 |
Classification | msc 60-00 |
Classification | msc 81-00 |
Classification | msc 83-00 |
Classification | msc 82-00 |
Classification | msc 55-00 |
Related topic | GeometricDistribution2 |
Related topic | StandardDeviation |
Related topic | Covariance |
Related topic | MeanSquareDeviation |