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Suppose I know how to generate independent Binomial Random Variables. How can I generate two random variables $X$ and $Y$ such that $$X\sim \text{Bin}(8,\dfrac{2}{3}),\quad Y\sim \text{Bin}(18,\dfrac{2}{3})\ \text{ and }\ \text{Corr}(X,Y)=0.5$$

I thought of trying to use the fact that $X$ and $Y-\rho X$ are independent where $\rho=Corr(X,Y)$ but I don't think that $X-\rho Y$ is Binomially distributed so I cannot use this method. If this had worked then I would have generated two Binomial random variables, say $A$ and $B$, then set $X=A$ and $Y-\rho X=B$ i.e. $Y=B+\rho A$ and hence I would have got the pair $(X,Y)$. But I cannot do this as $Y-\rho X$ is not Binomially distributed.

Any hint will be appreciated on how to proceed.

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  • $\begingroup$ Actually this problem came in a semester exam, so it is not Homework, but you can call it self-study I guess. Added the tag. $\endgroup$ Commented Apr 16, 2015 at 16:11

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You cannot use the linear representation of the correlation in discrete support distributions.

In the special case of the Binomial distribution, the representation $$X=\sum_{i=1}^8 \delta_i\quad Y=\sum_{i=1}^{18} \gamma_i\quad \delta_i,\gamma_i\sim \text{B}(1,2/3)$$ can be exploited since $$\text{cov}(X,Y)=\sum_{i=1}^8\sum_{j=1}^{18}\text{cov}(\delta_i,\gamma_j)$$ If we choose some of the $\delta_i$'s to be equal to some of the $\gamma_j$'s, and independently generated otherwise, we obtain $$\text{cov}(X,Y)=\sum_{i=1}^8\sum_{j=1}^{18}\mathbb{I}(\delta_i:=\gamma_j)\text{var}(\gamma_j)$$ where the notation $\mathbb{I}(\delta_i:=\gamma_j)$ indicates that $\delta_i$ is chosen identical to $\gamma_j$ rather than generated as a Bernoulli $\text{B}(1,2/3)$.

Since the constraint is$$\text{cov}(X,Y)=0.5\times\sqrt{8\times 18}\times\frac{2}{3}\times\frac{1}{3}$$we have to solve $$\sum_{i=1}^8\sum_{j=1}^{18}\mathbb{I}(\delta_i:=\gamma_j)=0.5\times\sqrt{8\times 18}=6$$This means that if we pick 6 of the 8 $\delta_i$'s equal to 6 of the 18 $\gamma_j$'s we should get this correlation of 0.5.

The implementation goes as follows:

  1. Generate $Z\sim\text{B}(6,2/3)$, $Y_1\sim\text{B}(12,2/3)$, $X_1\sim\text{B}(2,2/3)$;
  2. Takes $X=Z+Z_1$ and $Y=Z+Y_1$

We can check this result with an R simulation

> z=rbinom(10^8,6,.66)
> y=z+rbinom(10^8,12,.66)
> x=z+rbinom(10^8,2,.66)
cor(x,y)
> cor(x,y)
[1] 0.5000539

Comment

This is a rather artificial solution to the problem in that it only works because $8\times 18$ is a perfect square and because $\text{cor}(X,Y)\times\sqrt{8\times 18}$ is an integer. For other acceptable correlations, randomisation would be necessary, i.e. $\mathbb{I}(\delta_i:=\gamma_j)$ would be zero or one with some probability $\varrho$.

Addendum

The problem was proposed and solved years ago on Stack Overflow with the same idea of sharing Bernoullis.

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    $\begingroup$ +1. You don't need that $8\times 18$ be a square. The conditions for Cor$(X,Y)=\rho$ to have a solution (via this method) for $X\sim$ Binomial$(n,p)$ and $Y\sim$ Binomial$(m,q)$ are (1) $p=q$ and (2) $0\le \rho\sqrt{mn}\le \min(m,n)$ is an integer. For certain negative $\rho$, using a Multinomial distribution gives a solution. A more general--but more difficult approach--would use copulas. $\endgroup$ Commented Apr 16, 2015 at 15:42
  • $\begingroup$ @whuber: for the negative correlation, I first thought of using $1-\gamma_j$ but it obviously does not work. Could you expand on the generic solution? (I also thought of copulas, but calibrating copulas to reach the right correlation is a nasty business, isn't it?! $\endgroup$ Commented Apr 16, 2015 at 16:03
  • $\begingroup$ Xi'an, I would like to ask you whether the method you used is a standard one. This is because I searched a lot on the internet and could not find anything. $\endgroup$ Commented Apr 16, 2015 at 16:10
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    $\begingroup$ I agree with you--I don't want to work with those copulas! But they at least show that solutions ought to exist (within certain bounds on $\rho$, dependent on the other parameters) in the most general setting. It would be interesting to find out whether simpler constructions, such the one you give here, could be brought to bear to handle cases where $p\ne q$ or $\rho \lt 0$. Yedaynara: the method of splitting two variables $X,Y$ into $X^\prime+Z, Y^\prime+Z$ is standard for any parametric family closed under addition; and that's all that's going on here. $\endgroup$ Commented Apr 16, 2015 at 16:12
  • $\begingroup$ @yedaynara: I am surprised you could not find "anything" as I Googled "correlated Binomial simulation" and found immediately this post on Stack Overflow. $\endgroup$ Commented Apr 16, 2015 at 16:18

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