Appendix D: Function of Random Variables

Let $X=(x_1,\cdots,x_n)^T$ is an n-dimensional random vector with mean vector and covariance matrix:

$$M_x=E(X)=[\mu_1,\cdots,\mu_n]^T$$

$$\Sigma_x=E(X-M_x)(X-M_x)^T=E (XX^T)-M_x M_x^T =\left[ \begin{... .....&...\\ \sigma_{n1}^2 &...&\sigma_{nn}^2 \end{array} \right]$$

A linear transform of $X$ can be defined by an m by n matrix $A$ and the result $Y=AX$ is an m-dimensional random vector with the mean vector and covariance matrix:

$$M_y=E(Y)=E(AX)=A\;E(X)=A M_x$$

$$\Sigma_y = E(Y-M_y)(Y-M_y)^T =E(YY^T)-M_yM_y^T$$

$$= E(AX X^TA^T)-AM_x M_x^TA^T=A[E(XX^T)-M_xM_x^T]A^T =A\Sigma_x A^T$$

In particular, if $m=1$, $y=\sum_{j=1}^n a_j x_j$ is a linear combination of $(x_1,\cdots,x_n)$ and its mean is

$$\mu_y=E(y)=\sum_{j=1}^n a_j \mu_j$$

and its variance is

$$\sigma^2_y=A\Sigma_x A^T=[a_1,\cdots,a_n] \left[\begin{array... ...rray} \right]=\sum_{i=1}^n \sum_{j=1}^n a_i a_j \sigma_{ij}^2$$

If $(x_1,\cdots,x_n)$ are independent, i.e., $\sigma_{ij}=0$ for $i\ne j$, then

$$\sigma^2_y=\sum_{j=1}^n a_j^2 \sigma_j^2$$