Variance
Definition taught in Random variables and distributions, part of Probability
Variance
| Type | Definition |
| Taught in | Random variables and distributions |
| Part | Probability |
| Worked examples | 2 |
The variance of X is \operatorname{Var}(X)=\mathbb{E}(X-\mathbb{E} X)^2, when that expectation exists. Equivalently, \operatorname{Var}(X)=\mathbb{E} X^2-(\mathbb{E} X)^2. The standard deviation is the square root of the variance.
Examples
Variance on the unit interval
Question page: Variance on the unit interval
Let X be uniform on [0,1]. Calculate \operatorname{Var}(X).
By symmetry \mathbb{E} X=1/2. And \mathbb{E} X^2=\int_0^1 x^2\,\,\mathrm{d} x=1/3, so \operatorname{Var}(X)=1/3-1/4=1/12.
A game on the sum of two dice
Question page: A game on the sum of two dice
Two fair dice are thrown and Z is the sum of the faces.
Give the probability mass function of Z.
Calculate \operatorname{Var}(Z).
A player is paid (Z-7)^2 and pays 5 to play. Find the expected profit from 100 independent plays.
- Of the 36 equally likely ordered pairs, 6-|z-7| have sum z, so
\mathbb{P}(Z=z)=\frac{6-|z-7|}{36},\qquad z=2,3,\ldots,12.
Z is the sum of two independent faces, each with mean 7/2 and variance 35/12, so \mathbb{E} Z=7 and \operatorname{Var}(Z)=2\times35/12=35/6\approx5.83.
The payout is (Z-7)^2, whose expectation is \operatorname{Var}(Z)=35/6. The expected profit per play is 35/6-5=5/6, so over 100 independent plays it is 100\times5/6\approx83.3.
Used in
- Taught in Random variables and distributions, where the derivation and examples are.
- All 27 questions on Random variables and distributions
See also
Expectation · Random variable · Cumulative distribution function · Discrete distribution · Continuous distribution · Expectation of a function · Probability generating function · Moment generating function