# Thread: covariance, a normal vector (X,Y) and the distribution of Z = X - 2Y + 1

1. ## covariance, a normal vector (X,Y) and the distribution of Z = X - 2Y + 1

hi, I am preparing for a final in my least favorite class and could use a little help with this preparation problem.

Initially X and Y were normal with given mean and variance, and it asked for the distribution of the same Z = h(X,Y). I know that's a linear combination and was able to calculate the distribution no problem.

Now I am a little stuck on the next part,
Assume now that (X,Y) is a normal vector with covariance Cov(X,Y) = 4. What is the distribution of Z = X - 2Y + 1? Compute P( Z > 5).

I can get the second part if I am able to find the distribution of Z, which is where I get stumped.

Does the information in the beginning about the normal dist. of X, Y help? All I can see in my notes about this is

Cov(X,Y) = E(XY) - ux*uy where ux is the mean of x, same for y

thanks for any help at all!

2. Z = X - 2Y + 1

$\displaystyle V(aX+bY +c)=a^2V(X)+b^2V(Y)+2abCOV(X,Y)$

so $\displaystyle V(Z)=V(X)+4V(Y)-4COV(X,Y)$