Variance

Let $X$ be a (scalar) random variable taking value in some set $\mathcal{S}$. Then,

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

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

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

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

See also: Jensen’s inequality