Definition

Let XX be a (scalar) random variable taking value in some set 𝒮\mathcal{S}. Then, the variance is

Var[X]=𝔼[(X−𝔼[X])2]=𝔼[X2]−𝔼[X]2≤𝔼[X2]\mathrm{Var}[X]=\mathbb{E}[(X-\mathbb{E}[X])^2]=\mathbb{E}[X^2]-\mathbb{E}[X]^2 \leq \mathbb{E}[X^2]

Proposition

Var[X+Y]=Var[X]+Var[Y]+2Cov[X,Y]\mathrm{Var}[X+Y]=\mathrm{Var}[X]+\mathrm{Var}[Y]+2\mathrm{Cov}[X,Y]

If XX and YY are independent, then covariance Cov[X,Y]=0\mathrm{Cov}[X,Y]=0. (under this condition there is linearity of variance)

Notes

References:

  1. http://theanalysisofdata.com/probability/2_3.html
  2. https://www.kellogg.northwestern.edu/faculty/weber/decs-433/Notes_4_Random_variability.pdf
  3. https://cs229.stanford.edu/section/cs229-prob.pdf
  4. https://stats.stackexchange.com/questions/184998/the-linearity-of-variance
  5. https://math.stackexchange.com/questions/1708266/why-square-a-constant-when-determining-variance-of-a-random-variable

See also: Jensen’s inequality