Combining two dependant discrete random variables

#1
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) = ?
 
Last edited:

BGM

TS Contributor
#2
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 \( \rho \in (-1, 1) \)

Note that, as you displayed in a talbe, the number of support points of the joint probability mass function \( = 2 \times 3 = 6 \) (6 unknowns)

Also, from the table you have \( (2 - 1)(3 - 1) = 2 \) 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,

\( E[XY] = \rho SD[X]SD[Y] + E[X]E[Y] \)

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