Question

what is the correlation coefficient between x and x-5?​

Answer 1

Answer:

x is the correlation coefficient

Answer 2

Answer:

The correlation coefficient between x and x-5 is 1.

Step-by-step explanation:

The correlation coefficient between two terms x and y can be represented by r(x,y).

Therefore, we have to find r(x,x-5).

We know that,

$r(x,y) = \frac{cov(x,y)}{\sqrt{var(x)}\sqrt{var(y)} }$

where cov(x,y) represents the co-variance of x and y, and var(x) represents the variance of x.

Therefore, for y=x-5, we get:

$r(x,x-5) = \frac{cov(x,x-5)}{\sqrt{var(x)}\sqrt{var(x-5)} }$

Also,

$cov(x,x-5)=cov(x,x)\\cov(x,x-5)=var(x)$

Substituting above value of cov(x,x-5), we get:

$r(x,x-5) = \frac{var(x)}{\sqrt{var(x)}\sqrt{var(x-5)} }$

It is also known that:

$var(x-5)=var(x)$

Therefore,

$r(x,x-5) = \frac{var(x)}{\sqrt{var(x)}\sqrt{var(x)} }\\r(x,x-5)=1$

Thus, the correlation coefficient between x and x-5 is 1.

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