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I was wondering how to (or if it is even possible) find the continuous joint distribution between two random variables $x$ and $y$ when you know the continuous marginal density distributions of both $x$ and $y$, and we know there is a correlation between the two random variables, i.e., ${\rm Cor}(x,y)$ = $\rho$ which is of unknown magnitude?

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From Sklar's Theorem, it follows that you can construct the joint distribution using a copula:

$$H(x,y) = C(F(x),G(y)).$$

So, you need two ingredients: the marginal distributions $(F,G)$, and the copula $C$. You mentioned that you know the marginals, so this ingredient is done. Now, you need information to construct the copula. So, if you cannot come up with enough information to select/estimate/guess/divine the copula, then you cannot construct the joint distribution.

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Supposed that in Excel you create z-score of random variable P and Q, using a built-in function

=norminv(rand(),0,1)

If ρ is specified, then

X = P and Y = P*ρ + Q*sqrt(1-ρ^2)

The generated X and Y will distributed as normal distribution with zero mean and sd 1 with correlation coefficient = ρ.

I saw this formula a long time ago but could not find the reference now. Luckily, it works.

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