How can we derivate with a constant in trying to prove that correlation coefficient is between $-1$ and $1$?

Question

So I found this guy on Youtube who helped me understand how we prove that correlation coefficient is between $-1$ and $1$ but I didn't understand one part, that is the part when he derives equation with constant. Here is the proof (second picture) https://i.stack.imgur.com/PyeFX.jpg and proof video https://youtu.be/fTpTKRpMcFk?t=183

Answer 1

We suppose we have random variables $X$ and $Y$ with known distributions.

Then $\operatorname{Var}[2X+Y]$ is the variance of a variable $2X + Y.$ The variance is a real number.

Also $\operatorname{Var}[3X+Y]$ is the variance of another variable; that variance also is a real number.

Similarly $\operatorname{Var}[-17X+Y].$

In fact, you can take any real number $a$, put it in the expression

$$ \operatorname{Var}[aX+Y], $$

and out comes a real number, which happens to be the variance of the random variable you've just put together.

Or in other words we have a function $f$ such that for any real number $a,$

$$ f(a) = \operatorname{Var}[aX+Y]. $$

In fact it is a differentiable function, so you can take its derivative with respect to $a.$

We know that

$$ f(a) = a^2 \operatorname{Var}[X] + 2a\operatorname{Covar}[X,Y] + \operatorname{Var}[Y]. $$

And since $X$ and $Y$ are already known distributions that don't depend on $a,$ the expressions $\operatorname{Var}[X]$, $\operatorname{Covar}[X,Y],$ and $\operatorname{Var}[Y]$, which are determined just by those known distributions, are constants.