In: Finance
y=a+bx
b <0
prove that x and y are perfectly negatively correlated
Before we answer this question, you need to understand how square root function works.
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) = ρxyσxσy
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) = ρxyσxσy
We know, b < 0, σx2 > 0, hence only way the above equation can be zero if
which proves that x and y are perfectly negatively correlated.