Using the bilinearity of the covariance, we have cov(X,U)=cov(X,2X-3Y)=2cov(X,X)-3cov(X,Y)Let E(X) = 20, E(Y) = 10, Var(X) = 4 and Var(Y) = 1 and p=0.5 Additionally assume they are independent Find the covariance and correlation between X and U = 2X - 3Y.
Since X and Y are independent, cov(X,Y)=0
You're done for this one
As for the correlation, you'll need Var(U), which is Var(2X-3Y)=4Var(X)+9Var(Y) (because they're independent)
You're done for this one too
For the distribution of X1/X2, let's do it a smart wayAnd last question. Let X1 and X2 have both lognormal LN(0,1) and are both independent. Find the distribution of: X1/X2 and P(X1 > 2X2)
Since they're lognormal, there exist Y1 and Y2 both N(0,1) such that X1=exp(Y1) and X2=exp(Y2), and which are independent.
For further convenience, we can actually let X2=exp(-Y2) (it doesn't change much, because -N(0,1)=N(0,1))
Since Y1 and Y2 are independent, Y1+Y2 follows a normal distribution (0,2)
So this is a lognormal (0,2)
As for P(X1>2X2), note that it equals P(X1/X2>2)... or if you want to relate to a normal distribution, it equals P(N(0,2)>ln(2)).
If something I wrote is unclear, please tell me. It's late, and I was a bit lazy to explain very clearly