# Thread: Combining two dependant discrete random variables

1. ## Combining two dependant discrete random variables

Hi,
I’m looking for a way to combine two discrete random variables (which I have as probability distributions). The combination should be the product (or other operation) of the two variables.
This would be easy if they were independent, but they’re not. There is a known correlation between the variables.

Question: how to combine two discrete random variables with correlation?
Given: The marginal probabilities of the two variables & a correlation function
Result: either the individual probabilities in a probability table or the complete probability distribution of the combination.

Simple example:
Variables A and B are the distributions:
PA(a=1, 4) = [0.75, 0.25]
PB(b=4, 8, 10) = [0.25, 0.25, 0.5]

Their joint probability function is shown in their joint probability table and joint value table:
P B=4 8 10
A=1 ? ? ? 0.75
4 ? ? ? 0.25
0.25 0.25 0.5 1

value B=4 8 10
A=1 4 8 10
4 16 32 40

(tables are clearer in attached file)

The correlation between the two variables is: b = 10 – 2/3*a

P(A*B)(4, 8, 10, 16, 32, 40) = ?

2. ## Re: Combining two dependant discrete random variables

Originally Posted by simcc
Hi,
I’m looking for a way to combine two discrete random variables (which I have as probability distributions). The combination should be the product (or other operation) of the two variables.
This would be easy if they were independent, but they’re not. There is a known correlation between the variables.

Question: how to combine two discrete random variables with correlation?
Given: The marginal probabilities of the two variables & a correlation function
Result: either the individual probabilities in a probability table or the complete probability distribution of the combination.

Simple example:
Variables A and B are the distributions:
PA(a=1, 4) = [0.75, 0.25]
PB(b=4, 8, 10) = [0.25, 0.25, 0.5]

Their joint probability function is shown in their joint probability table and joint value table:
P B=4 8 10
A=1 ? ? ? 0.75
4 ? ? ? 0.25
0.25 0.25 0.5 1

value B=4 8 10
A=1 4 8 10
4 16 32 40

(tables are clearer in attached file)

The correlation between the two variables is: b = 10 – 2/3*a

P(A*B)(4, 8, 10, 16, 32, 40) = ?
What you have called a correlation is a functional relation between the values of the random variables, the correlation is a number $\displaystyle E((a-\overline{a})(b-\overline{b}))$. The relation is impossible in the form you present it.

Now assuming that you have the actual correlation:

Since two of the cells can be assigned arbitrarily within the constraints of the marginals the joint distribution has two degrees of freedom.

The correlation gives you one equation in the two variables that represent the degrees of freedom.

Usually a single equation in two real variables does not have a unique solution, and that is very likely the case here (it is possible that there is a unique solution but you won't know untill you so the calculations).

You almost certainly will need an additional constraint to get a unique solution.

CB