Question

If X and Y are independent random variables then the covariance between X+Y and X-Y is​

Answer 1

Answer:

search on Google thanks for free points

Answer 2

If X and Y are independent random variables then the covariance between X+Y and X-Y is​ zero.

Given:

X and Y are independent random variables.

To find:

The covariance between X+Y and X-Y.

Formulas used:

• $\ope{Cov}(X+Y,X-Y)=\mathbb{E}[(X+Y)(X-Y)]-\mathbb{E}[X+Y]\mathbb{E}[X-Y]$
• $\mathbb{E}[X]+\mathbb{E}[Y]=0$
• $\operator{Var}[X]=\mathbb{E}\left[ {{X}^{2}} \right]-\mathbb{E}{{[X]}^{2}}$

Step-by-step explanation

$\ope{Cov}(X+Y,X-Y)=\mathbb{E}[(X+Y)(X-Y)]-\mathbb{E}[X+Y]\mathbb{E}[X-Y]$

$=\mathbb{E}[(X+Y)(X-Y)]-(\mathbb{E}[X]+\mathbb{E}[Y])\mathbb{E}[X-Y])$

Since $\mathbb{E}[X]+\mathbb{E}[Y]=0$

$=\mathbb{E}[(X+Y)(X-Y)]-(0)\mathbb{E}[X-Y])$

$=\mathbb{E}[(X+Y)(X-Y)]$

$=\mathbb{E}\left[ {{X}^{2}}-{{Y}^{2}} \right]$

$=\mathbb{E}\left[ {{X}^{2}} \right]-\mathbb{E}\left[ {{Y}^{2}} \right]$

Since $\operator{Var}[X]=\mathbb{E}\left[ {{X}^{2}} \right]-\mathbb{E}{{[X]}^{2}}$

$=\operator{Var}[X]+\mathbb{E}{{[X]}^{2}}-\left( \operator{Var}[Y]+E{{[Y]}^{2}} \right)$

$=1+0-1-0$

$=0$

Hence, the covariance between X+Y and X-Y is​ zero.