# correlated binary variables

#### torkel

##### New Member
I have the following problem:
let X = {x_i}, and Y = {y_i} be two vectors of binary variables with n elements (taking the value 0 or 1 with some probability). X and Y have the same distribution.

I'm interested in the following quantity

Pr( Sum_{i=1}^n x_i *y_i )

That is, the probability distribution the 'overlap' of the two vectors (sum from i=0 to n). I have managed to find the distribution for this if all the x_i are not correlated. Its just a simple binomial. But want I really to calculate is the probability dist. of the overlap when the x_i (and y_i) are correlated in the following way.

E[ x_i x_j ] = E[ y_i y_j ] = const * |i-j|^-alfa

where alfa is a positive number.

Is this possible? As I do not have a background in probability theory, I hope the question make sense. I would highly appreciate any help or suggestions.

--
jon

#### Dragan

##### Super Moderator
I have the following problem:
let X = {x_i}, and Y = {y_i} be two vectors of binary variables with n elements (taking the value 0 or 1 with some probability). X and Y have the same distribution.

I'm interested in the following quantity

Pr( Sum_{i=1}^n x_i *y_i )

That is, the probability distribution the 'overlap' of the two vectors (sum from i=0 to n). I have managed to find the distribution for this if all the x_i are not correlated. Its just a simple binomial. But want I really to calculate is the probability dist. of the overlap when the x_i (and y_i) are correlated in the following way.

E[ x_i x_j ] = E[ y_i y_j ] = const * |i-j|^-alfa

where alfa is a positive number.

Is this possible? As I do not have a background in probability theory, I hope the question make sense. I would highly appreciate any help or suggestions.

--
jon
I think the answer you are looking for is in this short article:

Oman, S. D. & Zucker, D. M. (2001) Modelling and Generating Correlated Binary Variables, Biometrika, 88, 287-290.

#### torkel

##### New Member
I think the answer you are looking for is in this short article:

Oman, S. D. & Zucker, D. M. (2001) Modelling and Generating Correlated Binary Variables, Biometrika, 88, 287-290.
Thanks for the answear Dragan. But I don't see how this helps me.
They seem to discuss the case when the correlationfunction c_ij = alfa^|i-j| (alfa=gamma in the article) but I'm interested in the case when c_ij = |i-j|^alfa (the article makes a big deal of the case when the marginal probability pr(x_i=1)=p_i, that is dependent on i, in my case the marginal probability pr(x_i=1)=p that is the same for every i). That said i do not follow all the arguments in the article. Maybe i misunderstood something.

(btw. how do i write LaTeX formulas in this forum)