suppose $(X,Y)\sim\mathcal{N}(0,0,1,1,\rho)$. how can find distribution $Z=XY$ please explain completely
finding distribution $XY$ in bivariate normal distribution
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statistics
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0If $\rho=1$, 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. – 2012-03-11
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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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2Have 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 $Z$. – 2012-03-11