The Covariance function computes the covariance of two data samples or the covariance of multiple data samples in a Matrix.

The first parameter can be a data sample (given as e.g. a Vector), a Matrix data sample, a random variable, or an algebraic expression involving random variables (see Student[Statistics][RandomVariable]).

If the option inert is not included or is specified to be inert=false, then the function will return the actual value of the result. If inert or inert=true is specified, then the function will return the formula of evaluating the actual value.

If a computation involves floating point data or the option numeric = true or numeric is specified, then the result is a floating point number. Otherwise, the result is an exact expression.

$\mathrm{with}\left(\mathrm{Student}\left[\mathrm{Statistics}\right]\right)\:$

$A≔\left[1\,4\,5\,2\right]$

$A≔\left[1\,4\,5\,2\right]$

$B≔\left[2\,\mathrm{\pi }\,\mathrm{sqrt}\left(2\right)\,4\right]$

$B≔\left[2\,\mathrm{\pi }\,\sqrt{2}\,4\right]$

$\mathrm{Covariance}\left(A\,B\right)$

$-\frac{8}{3}+\frac{\mathrm{\pi }}{3}+\frac{2\sqrt{2}}{3}$

$\mathrm{Covariance}\left(A\,B\,\mathrm{numeric}\right)$

$\mathrm{-0.676660073888006}$

$U≔⟨\mathrm{seq}\left(57..77\right)\,\mathrm{undefined}⟩$

$V≔⟨\mathrm{seq}\left(\sin \left(i\right)\,i=57..77\right)\,\mathrm{undefined}⟩$

$\mathrm{Covariance}\left(U\,V\right)$

$\mathrm{undefined}$

$M≔\mathrm{Matrix}\left(\left[\left[1\,3\,10\right]\,\left[3\,4.1\,2\right]\,\left[10\,\mathrm{\pi }\,\mathrm{undefined}\right]\right]\right)$

$M≔\left[\begin{array}{ccc}1& 3& 10\\ 3& 4.1& 2\\ 10& \mathrm{\pi }& \mathrm{undefined}\end{array}\right]$

$\mathrm{Covariance}\left(M\right)$

$\left[\begin{array}{ccc}\frac{67}{3}& \mathrm{-0.539086257093885}& \mathrm{undefined}\\ \mathrm{-0.539086257093885}& 0.358098853533942& Float\left(\mathrm{undefined}\right)\\ \mathrm{undefined}& Float\left(\mathrm{undefined}\right)& \mathrm{undefined}\end{array}\right]$

$X≔\mathrm{NormalRandomVariable}\left(a\,b\right)\:$

$Y≔\mathrm{NormalRandomVariable}\left(c\,d\right)\:$

$\mathrm{Covariance}\left(X+Y\,X-Y\right)$

$a^2+b^2-c^2-d^2-\left(a+c\right)\left(a-c\right)$

$J≔\mathrm{PoissonRandomVariable}\left(\mathrm{\pi }\right)\:$

$K≔\mathrm{PoissonRandomVariable}\left(1\right)\:$

$\mathrm{Covariance}\left(JK\,K^2\,\mathrm{inert}\right)$

$\left(\sum _{\mathrm{\_t0}=0}^{\mathrm{\infty }}\frac{\left(\sum _{\mathrm{\_t}=0}^{\mathrm{\infty }}\frac{\mathrm{\_t}{\mathrm{\_t0}}^3{\mathrm{\pi }}^{\mathrm{\_t}}{\ⅇ}^{-\mathrm{\pi }}}{\mathrm{\_t}!}\right){\ⅇ}^{\mathrm{-1}}}{\mathrm{\_t0}!}\right)-\left(\sum _{\mathrm{\_t2}=0}^{\mathrm{\infty }}\frac{\left(\sum _{\mathrm{\_t1}=0}^{\mathrm{\infty }}\frac{\mathrm{\_t1}\mathrm{\_t2}{\mathrm{\pi }}^{\mathrm{\_t1}}{\ⅇ}^{-\mathrm{\pi }}}{\mathrm{\_t1}!}\right){\ⅇ}^{\mathrm{-1}}}{\mathrm{\_t2}!}\right)\left(\sum _{\mathrm{\_t3}=0}^{\mathrm{\infty }}\frac{{\mathrm{\_t3}}^2{\ⅇ}^{\mathrm{-1}}}{\mathrm{\_t3}!}\right)$

$\mathrm{evalf}\left(\mathrm{Covariance}\left(JK\,K^2\,\mathrm{inert}\right)\right)$

$9.424777962$

$\mathrm{Covariance}\left(JK\,K^2\,\mathrm{numeric}\right)$

$9.424777970$

Stuart, Alan, and Ord, Keith. Kendall's Advanced Theory of Statistics. 6th ed. London: Edward Arnold, 1998. Vol. 1: Distribution Theory.

The Student[Statistics][Covariance] command was introduced in Maple 18.

For more information on Maple 18 changes, see Updates in Maple 18.