Re: Joint Probability Question


lenkofman@xxxxxxxxx wrote:
>I just came across an interesting question and was wondering if some
>one could help me out. I have N binary random variables each with a
>50% chance of being 0 and 50% chance of being 1. The random variable
>are correlated and the correlation matrix is known. Is there an
>analytical way to compute the joint probabilities (e.g.
>P(x1=1,x2=1,x3=1,...xn=1) and all the other joint probablities)?
>

The given information isn't enough to specify the multivariate distribution.

Consider the following joint distribution

x1 x2 x3 prob
--------------
0 0 0 0.25
0 0 1 0.00
0 1 0 0.00
0 1 1 0.25
1 0 0 0.00
1 0 1 0.25
1 1 0 0.25
1 1 1 0.00

It's easy to compute that

P(x1=1)=0.5
P(x2=1)=0.5
P(x3=1)=0.5

Var(x1)=0.25
Var(x2)=0.25
Var(x3)=0.25

Cov(x1,x2)=0
Cov(x1,x3)=0
Cov(x2,x3)=0

Next, consider the distribution

x1 x2 x3 prob
---------------
0 0 0 0.125
0 0 1 0.125
0 1 0 0.125
0 1 1 0.125
1 0 0 0.125
1 0 1 0.125
1 1 0 0.125
1 1 1 0.125

Again, we have that

P(x1=1)=0.5
P(x2=1)=0.5
P(x3=1)=0.5

Var(x1)=0.25
Var(x2)=0.25
Var(x3)=0.25

Cov(x1,x2)=0
Cov(x1,x3)=0
Cov(x2,x3)=0

In one case, P(x1=1,x2=1,x3=1)=0.00, while in the other case it's
P(x1=1,x2=1,x3=1)=0.125.
--
Brian Borchers borchers@xxxxxxx
Department of Mathematics http://www.nmt.edu/~borchers/
New Mexico Tech Phone: 505-835-5813
Socorro, NM 87801 FAX: 505-835-5366
.

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