Linear Transformations and Combinations

Linear Transformations

Consider the following linear transformations of the variables $x$ and $y$:

$$v=a+bx\text{ and }w=c+dy$$

where $a,b,c,$ and $d$ are all known constants.

With Univariate data, we know:

$\bar{v}=a+b\bar{x}$ $\bar{w}=c+d\bar{x}$ $s_v^2=b^2{s}_x^2$ $s_w^2=d^2{s}_y^2$ $s_v=|b|s_x$ $s_w=|d|s_y$

With respect to the linear association between $v$ and $w$, we can show the following:

$$s_{vw}=bd{s}_{xy}$$ $$\begin{array}{r}{r}_{vw}=\{\begin{array}{l}{r}_{xy}\text{ if b and d are both positive or both negative}\\ -r_{xy}\text{ if b and d are opposite signs}\end{array}\end{array}$$

Linear Combinations

With two variables, one can also consider a linear combination of the variables $x$ and $y$ as follows:

$$v=k+ax+by$$

where $k,a,$ and $b$ are known constants.

We can then show the following:

Sample mean of $v$: $\bar{v}=k+a\bar{x}+b\bar{y}$

Sample variance of $v$:$s_v^2=a^2{s}_x^2+b^2{s}_y^2+2ab{s}_{xy}$

Sample standard deviation of $v$: $s_v=\sqrt{s_v^2}=\sqrt{a^2{s}_x^2+b^2{s}_y^2+2ab{s}_{xy}}$