# #StackBounty: #correlation #mathematical-statistics #simulation #copula If I have a vector of \$N\$ correlated probabilities. How can I t…

### Bounty: 300

My ultimate goal is to be able to have a way to generate a vector of size \$N\$ of correlated Bernoulli random variables. One way I am doing this is to use the Gaussian Coupla approach. However, the Gaussian Coupla approach just leaves me with a vector:

\$\$
(p_1, ldots, p_N) in [0,1]^N
\$\$

Suppose that I have generated \$(p_1, ldots, p_N)\$ such that the common correlation between them is \$rho\$. Now, how can I transform these into a new vector of \$0\$ or \$1\$’s? In other words, I would like:

\$\$
(X_1, ldots, X_N) in {0,1}^N
\$\$

but with the same correlation \$rho\$.

One approach I thought of was to assign a hard cutoff rule such that if \$p_i < 0.5\$, then let \$X_i = 0\$ and if \$p_i geq 0.5\$, then let \$X_i = 1\$.

This seems to work well in simulations in that it retains the correlation structure but it is very arbitrary to me what cutoff value should be chosen aside from \$0.5\$.

Another way is to treat each \$X_i\$ as a Bernoulli random variable with success probability \$p_i\$ and sample from it. However this approach seems to cause loss of correlation and instead of \$rho\$, I may get \$frac{rho}{2}\$ or \$frac{rho}{3}\$.

Does anyone have any thoughts or inputs into this? Thank you.

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