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

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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
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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 $Z$.

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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
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