Question

y=a+bx
b <0
prove that x and y are perfectly negatively correlated

Answer 1

Before we answer this question, you need to understand how square root function works.

1. If b is positive, then
2. If b is negative, then

It's quite intuitive.

In the second case, b = -5 and the square root is 5 = -b.

This is the way excel also behaves.

In our question, b < 0. So wherever we encounter , we will replace it by -b.

From the elementary definition of covariance between two variables,

Covariance (x, y) = ρ_xyσ_xσ_y

Please note that we have made use of the fact that: If b is negative, then

Covariance (x, y) , by first principle, is given by

Putting all these things together

Covariance (x, y) = ρ_xyσ_xσ_y

We know, b < 0, σ_x² > 0, hence only way the above equation can be zero if

which proves that x and y are perfectly negatively correlated.