Homework help - calculating ρ

#1
Hi everyone, I'm new to the forum and I would really appreciate some help with a homework problem. Sorry about possible language mistakes, my native language is Swedish...

Here is my problem: Let Y=Z-3X^2 where X and Z are normally distributed N(0,1). Simulate n=50 observations of (X,Y) and plot Y against X. Estimate ρ(X, Y) and comment the result. What is the true ρ in this case? Does the result indicate that X and Y are stochastic independent?

I have done the plot and has had Excel estimate r=-0,1416, but I cannot understand how to calculate the true value of ρ. Maybe someone can give me a hint?
 
#2
Ok, no reply. Anyway, here is how I'm thinking right now. Since ρ=cov(X,Y)/sqrt(var(X)*var(Y) I need to find these numbers.

Cov(X,Y)=E(XY)-μx*μy and since both μx and μy are 0 I want to find E(XY). E(XY)=E(X)*E(Z)-3E(X^3) but since both E(X), E(Y) and E(X^3) equals 0 my cov(X,Y becomes 0. Something must be wrong?
 

Dragan

Super Moderator
#3
Ok, no reply. Anyway, here is how I'm thinking right now. Since ρ=cov(X,Y)/sqrt(var(X)*var(Y) I need to find these numbers.

Cov(X,Y)=E(XY)-μx*μy and since both μx and μy are 0 I want to find E(XY). E(XY)=E(X)*E(Z)-3E(X^3) but since both E(X), E(Y) and E(X^3) equals 0 my cov(X,Y becomes 0. Something must be wrong?
Looks good to me i.e. ρ=0.

What you might want to do is generate, say, 1000000 data points (of X and Z) and confirm what you have computed analytically. It's fairly easy thing to do.