(adsbygoogle = window.adsbygoogle || []).push({}); [SOLVED] Help with variance sum + correlation coefficient formula

This is a worked example

The objective is to prove

[tex]-1 \leq \rho(X,Y) \leq 1[/tex]

Then the book uses this formula...

(2) [tex]0 \leq Var(\left \frac{X}{\sigma_x} + \frac{Y}{\sigma_y} \right)[/tex]

(3) [tex]= \frac{Var(X)}{{\sigma_x}^2} + \frac{Var(Y)}{{\sigma_y}^2} + \frac{2Cov(X,Y)}{\sigma_x \sigma_y}[/tex]

The question is, how does 2 lead to 3? Namely, how does [tex]Var(\frac{X}{\sigma_x} ) => \frac{Var(X)}{{\sigma_x}^2}[/tex]?

Also, how does one get the idea to use formula (2) to prove (1)? It doesn't seem like a natural step

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# Help with variance sum + correlation coefficient formula

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