Question

If two random variables x and y, are connected by relationship $2x + y = 3$, then $r_{xy} =$

• A. 1
• B. – 1
• C. – 2
• D. 3

Answer 1

Answer: (B)

– 1

Answer 2

Since $2x + y = 3$

$\therefore$ ${cov}(x,y) = \frac{1}{n}\sum_{i = 1}^{n}{x_{i}.y_{i} - \left( \frac{1}{n}\sum_{i = 1}^{n}x_{i} \right)\left( \frac{1}{n}\sum_{i = 1}^{n}y_{i} \right)}$; $\therefore$ $y - \bar{y} = - 2(x - \bar{x})$. So, $b_{yx} = - 2$

Also $x - \bar{x} = - \frac{1}{2}(y - \bar{y})$, $\therefore$ $b_{xy} = - \frac{1}{2}$

$\therefore$ $r_{xy}^{2} = b_{yx}.b_{xy} = ( - 2)\left( - \frac{1}{2} \right) = 1$ ⇒ $r_{xy} = - 1$.

($\because$both $b_{yx},b_{xy}$ are –ive)