Browse Questions

# The correlation between $x$ and $a-x$ is

$\begin {array} {1 1} (A)\;-1 & \quad (B)\;1 \\ (C)\;\large\frac{1}{2} & \quad (D)\;0 \end {array}$

Let $u = a-x$ and therefore
Var $(u)$ = Var $(a-x)$
$= (-1)^2$ var $(x)$
= var $(x) = \sigma^2$
cov $(x, a-x) =$ cov $(x, u)$
$= E \{ (x- \overline x )(u-\overline u) \}$
$E (x-\overline x )^2\: \: \: [ \because u- \overline u = -(x-\overline x ) ]$
= - var $(x)$
$= -\sigma^2$
Hence $r(x,u) = \large\frac{cov (x,u)}{\sqrt { var(x)var(u)}}$
$= \large\frac{-\sigma^2}{\sqrt{\sigma^2, \sigma^2}}$
$= -1$
Hence Ans : (A)