Help with variance sum + correlation coefficient formula

[Simfish]

[SOLVED] Help with variance sum + correlation coefficient formula

This is a worked example

The objective is to prove

$$-1 \leq \rho(X,Y) \leq 1$$

Then the book uses this formula...

(2) $$0 \leq Var(\left \frac{X}{\sigma_x} + \frac{Y}{\sigma_y} \right)$$

(3) $$= \frac{Var(X)}{{\sigma_x}^2} + \frac{Var(Y)}{{\sigma_y}^2} + \frac{2Cov(X,Y)}{\sigma_x \sigma_y}$$

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

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

 

[Simfish]

sorry, I edit my posts a lot - so somehow, edited posts on PF don't edit the tex code any longer once you edit the posts enough...

Namely, how does $$Var(\frac{X}{\sigma_x})$$ => $$\frac{Var(X)}{{\sigma_x}^2}$$?

 

[EnumaElish]

What is Var(aX), where a is constant?

 

[Simfish]

aVar(X)

holy crap
i never knew my attention lapses were that bad

 

[D H]

aVar(X)

That's not right. The variance of a one-dimensional random variable X is defined as $\text{Var}(X) = \text{E}[(X-\text{E}(X))^2]$. What does this mean in terms of scaling X by some quantity a?

 

[Simfish]

Oh, I see.

$$a^2 Var(X)$$

 

[D H]

Now onto the next question: Given two random variables $X$ and $Y$, what is $\text{Var}(X+Y)$? Apply the definition of $\text{Var}(X)$ to the new random variable $X+Y$.

 

[Simfish]

$$= {Var(X)} + {Var(Y)} + {2Cov(X,Y)}$$

but that's from memorization - I'll try to derive it now

 

[Simfish]

$$Var(X+Y)$$

$$= E[(X+Y)^2] - E[X+Y]^2$$

$$= E[X^2 + 2XY + Y^2] - E[X+Y]^2$$
$$= E[X^2] + 2E[XY] + E[Y^2] - (E[X] + E[Y])^2$$
$$= E[X^2] + 2E[XY] + E[Y^2] - E[X]^2 - E[Y]^2 - 2E[X]E[Y]$$
$$= (E[X^2]- E[X]^2) + (E[Y^2] - E[Y]^2) + (2E[XY]- 2E[X]E[Y])$$
$$= Var(X) + Var(Y) + 2Cov(X,Y)$$ IF dependent
IF independent, 2E[XY] = 2E[X]E[Y]

==
Okay, can someone please address my second question?
Also, how does one get the idea to use formula (2) to prove (1)? It doesn't seem like a natural step

 

[D H]

What is $\text{Var}\left(X/{{\sigma_x}}\right)$?

 

[Simfish]

$$\frac{Var(X)}{{\sigma_x}^2} = 1$$ since $$Var(X) = \sigma_x, Var(Y) = \sigma_y$$. I have the entire proof in the book - but the first step seems unnatural (how does one get the inspiration to use $$0 \leq Var(\left \frac{X}{\sigma_x} + \frac{Y}{\sigma_y} \right)$$ for proving that correlation coefficient has absolute magnitude <= 1?

 

[D H]

The variance of any random variable is tautologically non-negative. Look at the definition of variance.

 

[EnumaElish]

You mean $$Var(X) = \sigma_x^2, Var(Y) = \sigma_y^2$$

but the first step seems unnatural

The idea behind correlation is to standardize variables X and Y by dividing each by its standard deviation before finding their correlation.

 

[Simfish]

Okay I see. Thanks. :)