2

Let $X, Y, Z$ be discrete Bernoulli random variables with parameters $θ_{1}, θ_{2}, θ_{3}$, respectively, such that $0 <θ_{i} <1$, $i ∈ {1, 2, 3}$ Construct the class of all functions trivariate joint probability mass, adding the necessary parameters, and determine whether or not this class includes the case of independent random variables.

Hello, I need help to solve this exercise, I understand that it is analogous to the bivariate case, in which I built tables and found parameters and the class of probability mass functions that turned out to be $P: = \{ p_{X,Y} (x, y) : 0 <θ_{1} <1, 0 <θ_{2} <1$, $max \{θ_{1} + θ_{2} - 1, 0\} ≤ α ≤ min \{θ_{1}, θ_{2}\}\}$

The problem comes when I establish my systems of equations and I see that there is no value that I can use as a parameter, I attach the development that I have done.

Let

$p_{x}(x)= \left\{\begin{matrix} 1-θ_{1}, x=0\\ θ_{1}, x=1 \end{matrix}\right.$

$p_{y}(y)= \left\{\begin{matrix} 1-θ_{2}, y=0\\ θ_{2}, y=1 \end{matrix}\right.$

$p_{z}(z)= \left\{\begin{matrix} 1-θ_{3}, z=0\\ θ_{3}, z=1 \end{matrix}\right.$

We have that $RanX=RanY=RanZ=\{0,1\}$ then $Ran(X,Y,Z) \subset \{(0,0,0), (0,0,1), (0,1,0), (0,1,1),(1,0,0), (1,0,1),(1,1,0),(1,1,1)\} $

Then, I built the table from which the following system of equations is obtained

$k_{000}+k_{001}=q_{00}\\ k_{010}+k_{011}=q_{01}\\ k_{100}+k_{101}=q_{10}\\ k_{110}+k_{111}=q_{11}\\ k_{000}+k_{010}+k_{100}+k_{110}=1-θ_{3}\\ k_{001}+k_{011}+k_{101}+k_{111}=θ_{3}\\ k_{000}+k_{010}+k_{100}+k_{110}+k_{001}+k_{011}+k_{101}+k_{111}=1 $

but from the bivariate model we have to

$q_{00}+q_{01}+q_{10}+q{11}=1\\ q_{10} = θ_{1} − q_{11} q_{01} = θ_{2} − q_{11} q_{00} = 1 − θ_{1} − θ_{2} + q_{11}$

with which it results

$k_{000}=k_{011}+k_{101}+k_{111}-θ_{1}- θ_{2}+ q_{11}- θ_{3}+1\\ k_{001}=2θ_{2}+θ_{1} -k_{001}-k_{101}-k_{111}\\ k_{010}=θ_{2}-q_{11}-k_{001}\\ k_{011}=k_{011}\\ k_{100}=θ_{1}-q_{11}-k_{101}\\ k_{101}=k_{101}\\ k_{110}=q_{11}-k_{111}\\ k_{111}=k_{111}$

From here on, I don't know how to proceed to find the class, or to determine if the dependency / independence case exists. Also I can't identify the parameters.

Fernanda Gutierrez
  • 33

1 Answers1

2

You are given the marginal distributions (and only that), hence the independence case trivially exists: $P(Z_1,Z_2,Z_3) = P(Z_1)P(Z_2)P(Z_3)$

In general, the joint distribution has $2^3=8$ vaues, but the conditions $\sum p=1$ and the three marginals leave you with $4$ degrees of freedom. You might parametrize that in several ways (you could take $k_{111}$, $k_{011}$ $k_{101}$ $k_{110}$ in your notation).

Here I proposed another parametrization, which might be considered more elegant, in that the parameters are directly related with the moments, and where independence is immediately spotted.

leonbloy
  • 63,430