The variance is a measure of the spread of a distribution.
Definition
The population variance of a random variable X is given by:
.
The population variance is often denoted by the square of the greek letter σ, so Var(X) = σ2. The population standard deviation is the square root of the population variance and is denoted by σ.
Covariance
The covariance of two random variables, X and Y, is given by:
Cov(X,Y) = E(XY) - E(X)E(Y)
Properties
Let X and Y be random variables, and a and b be real numbers.
- Var(aX) = a2 Var(X)
- Var(X + b) = Var(X)
- Var(a X + b) = a2 Var(X)
- Var(X + Y) = Var(X) + Var(Y) + 2Cov(X,Y)
- Var(a X + b Y) = a2 Var(X) + b2 Var(Y) + 2ab Cov(X,Y)
Estimation
The variance and covariance can be estimated from a random sample. If we have a random sample, , the sample variance is given by:
.
If we have a random sample from the joint distribution of the random variables X and Y, , then the sample covariance is given by:
.