Tell me more ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.
up vote 1 down vote favorite

suppose $(X,Y)\sim\mathcal{N}(0,0,1,1,\rho)$. how can find distribution $$Z=XY$$ please explain completely

share|improve this question
4  
Still not found the accept button? – Did Mar 11 '12 at 20:22
If $\rho=1$, it's a chi-square distribution with $1$ degree of freedom. If $\rho=-1$, just multiply the chi-square random variable by $-1$. But for $\rho$ between $0$ and $1$, I'm not sure this is easy to do, unless maybe by numerical methods. – Michael Hardy Mar 11 '12 at 20:27

1 Answer

up vote 1 down vote accepted

Hint: Start by defining a map $(X,Y)\rightarrow (Z,V)$ by $$Z=XY$$ and $$V=X$$. Use a method of transformation to calculate the density of $(Z,V)$ and then integrate out to get the marginal distribution of $Z$.

share|improve this answer
1  
Have you tried this approach yourself? The inverse transformation $X = V, Y = Z/V$ worries me a bit since $X$ and $Y$ are correlated and you have to integrate the joint density to get the marginal density of $Z$. – Dilip Sarwate Mar 11 '12 at 21:36

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.