# Thread: Combining normal random variable question

1. ## Combining normal random variable question

I am stuck on this question...

I know for part a) that E(X-Y) = -1 and Var(X-Y) = 5, but I'm not sure where to go from here...

Is X-Y still a normal random variable? If so then I get Norm(-1,5) but I don't know where to go from there either.

For part b) I don't understand what the correlation coefficient is and so I don't know how to go about doing this.

All help will be much appreciated

2. Originally Posted by neild
I am stuck on this question...

I know for part a) that E(X-Y) = -1 and Var(X-Y) = 5, but I'm not sure where to go from here...

Is X-Y still a normal random variable? If so then I get Norm(-1,5) but I don't know where to go from there either.

For part b) I don't understand what the correlation coefficient is and so I don't know how to go about doing this.

All help will be much appreciated
The linear combination of two independent normal variates is well known to be a normal variate. See Normal distribution - Wikipedia, the free encyclopedia

3. $\displaystyle W=X-Y\sim N(-1,5)$

So $\displaystyle P(|W|>3) =P(W<-3)+P(W>3)$

$\displaystyle =P(Z< {-3+1\over\sqrt{5}}) +P(Z> {3+1\over\sqrt{5}})$

$\displaystyle =P(Z< {-3+1\over\sqrt{5}}) +1-P(Z< {3+1\over\sqrt{5}})$

$\displaystyle =1+\Phi({-2\over\sqrt{5}}) -\Phi({4\over\sqrt{5}})$

4. The correlation is the covariance divided by the two st deviations.
I'll only do the covariance...

$\displaystyle Cov(2X+Y,X-Y) = 2Cov(X,X)+Cov(Y,X)-2Cov(X,Y)-Cov(Y,Y)$

$\displaystyle = 2V(X)+0-0-V(Y)$

=-2, which is legal.