PROBABILITY 10
Variance and Standard Deviation
Variance measures the spread of a random variable around its expected value.
Variance
Let \(\mu=E[X]\). The variance is
\[\boxed{\operatorname{Var}(X)=E[(X-\mu)^2]}\]
Squaring prevents positive and negative deviations from cancelling.
1. Computational form
Expanding the square gives
\[\boxed{\operatorname{Var}(X)=E[X^2]-(E[X])^2}\]
This connects variance to the first and second moments.
2. Standard deviation
\[\boxed{\operatorname{SD}(X)=\sqrt{\operatorname{Var}(X)}}.\]Standard deviation has the same physical units as \(X\), whereas variance has squared units.
3. Constant random variable
If \(X=c\) with certainty,
\[\operatorname{Var}(X)=0.\]There is no uncertainty around the mean.
4. Scaling
For constants \(a,b\),
\[\operatorname{Var}(aX+b)=a^2\operatorname{Var}(X).\]Adding a constant shifts the distribution without changing its spread.
5. Sum of independent variables
If \(X\) and \(Y\) are independent,
\[\operatorname{Var}(X+Y)=\operatorname{Var}(X)+\operatorname{Var}(Y).\]Without independence a covariance term is required.
6. Mean versus uncertainty
\[\boxed{E[X]\;\text{describes centre},\qquad \operatorname{Var}(X)\;\text{describes spread}}\]
Two models can have the same expectation but very different variances and therefore very different risk.