Home page for accesible maths 7 Linear transformations 7.4 Variances of Linear Transformations 8 Bivariate Transformations

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7.5 Key definitions and Relationships

Let $(X,Y)$ be a bivariate rv and $\boldsymbol{X}=(X_{1},\dots,X_{n})^{t}$ and $\boldsymbol{Y}=(Y_{1},\dots,Y_{m})$ be vector rvs.

1.

The covariance of $X$ and $Y$ is ${\operatorname{\mathsf{Cov}}}\left[{X,Y}\right]=\operatorname{\mathsf{E}}\left% [{(X-\operatorname{\mathsf{E}}\left[{X}\right])(Y-\operatorname{\mathsf{E}}% \left[{Y}\right])}\right]=\operatorname{\mathsf{E}}\left[{XY}\right]-% \operatorname{\mathsf{E}}\left[{X}\right]\operatorname{\mathsf{E}}\left[{Y}\right]$ . ${\operatorname{\mathsf{Var}}}\left[{X}\right]={\operatorname{\mathsf{Cov}}}% \left[{X,X}\right]$ .
2.

Bilinearity: ${\operatorname{\mathsf{Cov}}}\left[{aX,Y}\right]=a{\operatorname{\mathsf{Cov}}% }\left[{X,Y}\right]$ and ${\operatorname{\mathsf{Cov}}}\left[{W+X,Y}\right]={\operatorname{\mathsf{Cov}}% }\left[{W,Y}\right]+{\operatorname{\mathsf{Cov}}}\left[{X,Y}\right]$
3.

The correlation between $X$ and $Y$ is ${\operatorname{\mathsf{Corr}}}\left[{X,Y}\right]={\operatorname{\mathsf{Cov}}}% \left[{X,Y}\right]/\sqrt{{\operatorname{\mathsf{Var}}}\left[{X}\right]{% \operatorname{\mathsf{Var}}}\left[{Y}\right]}$ .
4.

Vector/matrix forms: $\operatorname{\mathsf{E}}\left[{\boldsymbol{X}}\right]=(\operatorname{\mathsf{% E}}\left[{X_{1}}\right],\dots,\operatorname{\mathsf{E}}\left[{X_{n}}\right])^{t}$ . ${\operatorname{\mathsf{Var}}}\left[{\boldsymbol{X}}\right]$ is the matrix with $(i,j)$ -th element ${\operatorname{\mathsf{Cov}}}\left[{X_{i},X_{j}}\right]$ .
5.

Summaries for linear transformations: $\operatorname{\mathsf{E}}\left[{A\boldsymbol{X}}\right]=A\operatorname{\mathsf% {E}}\left[{\boldsymbol{X}}\right]$ and ${\operatorname{\mathsf{Var}}}\left[{A\boldsymbol{X}}\right]=A{\operatorname{% \mathsf{Var}}}\left[{\boldsymbol{X}}\right]A^{\prime}$ , so $\operatorname{\mathsf{E}}\left[{\boldsymbol{a}^{\prime}\boldsymbol{X}}\right]=% \boldsymbol{a}^{\prime}\operatorname{\mathsf{E}}\left[{X}\right]$ , ${\operatorname{\mathsf{Var}}}\left[{\boldsymbol{a}^{\prime}\boldsymbol{X}}% \right]=\boldsymbol{a}^{\prime}{\operatorname{\mathsf{Var}}}\left[{\boldsymbol% {X}}\right]\boldsymbol{a}$ , and ${\operatorname{\mathsf{Cov}}}\left[{\boldsymbol{a}_{1}^{\prime}\boldsymbol{X},% \boldsymbol{a}_{2}^{\prime}\boldsymbol{X}}\right]=\boldsymbol{a}_{1}^{\prime}{% \operatorname{\mathsf{Var}}}\left[{\boldsymbol{X}}\right]\boldsymbol{a}_{2}$ .