Determine the type of the correlation

Question

If the equation of the regression line of y on x is $$y=3-x$$ Then the correlation between x and y is // (inverse , perfect inverse )? Does the coeffecient of x means the correlation in perfect as it equals -1 ?.

Answer 1

$$y=\frac{cov(X, Y)}{D^2X}(x-\mathbb{E}X)+\mathbb{E}Y $$ Coefficient $-1$ means the covariance of $X$ and $Y$ is negative, so that the corelation of $X$ and $Y$ is negative, in other words, we have inverse correlation. It's not necessarily (at least it doesn't look so) perfect inverse correlation.

Answer 2

The correlation would be negative because as $x$ increases, $y$ decreases and vice versa. A positive correlation would be something like $y = 3+x$. $x$ and $y$ would be inversely correlated for something like $y = 1/x$.

Answer 3

If $X$ and $Y$ are random variables, then

$$ {\rm Cov}[X, Y] = {\rm Cov}[X, 3 - X] = {\rm Cov}[X , -X] = -{\rm Cov}[X,X] = -\sigma_X^2 $$

So that the correlation coefficient is

$$ \rho_{X,Y} = \frac{{\rm Cov}[X, Y]}{\sigma_X \sigma_Y} = -\frac{\sigma_X^2}{\sigma_X \sigma_X} = -1 $$