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The covariance measures the average product of two centered deviations. For real square-integrable random variables $X$ and $Y$,Deviations with the same sign contribute positively; deviations with opposite signs contribute negatively. Their magnitudes and probabilities determine the net result. Taking $Y=X$ recovers the variance. The definitions and algebraic properties are presented in Orloff and Bloom, MIT 18.05, Covariance and Correlation, sections 2–2.1.

For a worked example, let $X,Z$ be independent random variables, each equally likely to be $-1$, $0$ or $1$, and put $Y=2X+Z$. All nine pairs $(X,Z)$ have probability $1/9$. The expected values are zero, while linearity of covariance and the vanishing covariance of independent random variables giveThe shared $X$ term makes large $X$ accompany large $Y$ on average. Replacing $2X$ by $-2X$ reverses the covariance to $-4/3$ without changing either variance.

The covariance is useful whenever deviations are added or subtracted. Expanding the square givesFor the worked model, these are $20/3$ and $4/3$. Positive covariance increases variation in the sum while reducing variation in the difference: the same shared fluctuation reinforces one combination and partly cancels in the other. The identity applies beyond independent random variables.

The size of a covariance depends on the measurement scale. For constants $a,b,c,d$, linearity of covariance gives $\mathrm{Cov}(aX+b,cY+d)=ac\mathrm{Cov}(X,Y)$. Adding offsets has no effect; rescaling either measurement rescales the covariance. The Pearson correlation coefficient divides by both standard deviations to describe the standardized linear relationship.

For any finite list $t_1,\dots ,t_k\in [0,1]$, the random vector $(U_{t_1},\dots ,U_{t_k},B_1)$ has a multivariate normal distribution, since it is a linear image of a vector of Brownian motion coordinates. The Brownian covariance kernel and linearity of covariance giveThe independence of uncorrelated jointly normal variables implies that $(U_{t_1},\dots ,U_{t_k})$ is independent of $B_1$. Extending from finite-coordinate cylinder events by independence extended from generating pi-systems provesThus the Brownian bridge is independent of its endpoint. Its covariance kernel is $\mathrm{Cov}(U_s,U_t)=\min (s,t)-st$, confirming that $U$ is the standard Brownian bridge. This is the Brownian bridge independence from its endpoint property.

For distinct sibling supernovae $s$ and $s^{\prime }$ in the same galaxy, the shared fluctuation is the only common random term. Independence and linearity of covariance therefore giveTheir correlation coefficient isThis is the intraclass correlation coefficient of the galaxy-level random intercept model.