# 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

The correlation between the two variables is: b = 10 – 2/3*a
I am not sure what the expression is on the RHS, but I assume that you are given the correlation coefficient of the two random variables to be a fixed constant

Note that, as you displayed in a talbe, the number of support points of the joint probability mass function (6 unknowns)

Also, from the table you have degrees of freedom, i.e. you need to have 2 conditions to solve the joint. The four known conditions come from the fact that the sum of row/columns are equal to the marginal, and one of the five equations are redundant.

From the definition of correlation,

so you have the 5th equation. But you lack the 6th one to uniquely solve the table.

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