How can I draw a distribution table for, say, 4 Random Variables?

Question

I know that a bi-variate distribution table would be a matrix which is very easy to draw/represent.

1. Suppose, we roll 4 six-sided dies simultaneously, and we want to draw/represent a multivariate distribution table for that experiment.

2. What if all are not dies? Say, we (a) roll a die, (b) flip a coin, (c) pick a random integer from 1 to 10, and (d) draw an ace card. Draw a distribution table for this experiment.

and, so on.

How can we do that?

Answer 1

If you want the pdf for the sum of the four dice you have $21$ possible outcomes ($4, 5, ...,24$),

Then you have to calculate the number of favorable outcomes for each sum. This can be done the the generating function:

$$(x^6+x^5+x^4+x^3+x^2+x^1)^4$$

Now you expand this expression. This can be done with the help of a calculator:

$$=1\cdot{{x}^{24}}+4\cdot {{x}^{23}}+10\cdot {{x}^{22}}+20\cdot {{x}^{21}}+35\cdot {{x}^{20}}+56\cdot {{x}^{19}}+80\cdot {{x}^{18}}\\+104\cdot {{x}^{17}}+125\cdot {{x}^{16}}+140\cdot {{x}^{15}}+146\cdot {{x}^{14}}+140\cdot {{x}^{13}}+125\cdot {{x}^{12}}+104\cdot {{x}^{11}}\\+80\cdot {{x}^{10}}+56\cdot {{x}^{9}}+35\cdot {{x}^{8}}+20\cdot {{x}^{7}}+10\cdot {{x}^{6}}+4\cdot {{x}^{5}}+1\cdot{{x}^{4}}$$

Let $Y_4=X_1+X_2+X_3+X_4$. Now you take the coefficients as the number of the favorable outcomes. The number of all possible outcomes is $6^4=1296$ Therefore the pdf of $Y_4$ is

\begin{array}{|c|c|c|c|c|} \hline y & 4 &5&6&7&8&\ldots&23&24 \\ \hline P(Y_4=y) &\frac{1}{1296}&\frac{4}{1296}&\frac{10}{1296}&\frac{20}{1296}&\frac{35}{1296}&\ldots&\frac{4}{1296}&\frac{1}{1296} \\ \hline \end{array}

In this case you can write it as an univariate distribution.

Answer 2

To display the joint distribution of discrete random variables in tabular form you'll want a separate entry for each possible combination of values for your random variables. When you're interested in two variables you can display every combination nicely in a two-dimensional table, with rows representing values of one variable and columns representing values of the other variable. For the three-variables analog you'll have trouble rendering the distribution as a 3-d matrix unless you are skilled in three-d drawing. (Or you might be able to display a 3-d table as a series of separate rectangular "sheets", one per "level" of the first variable.)

Once you get beyond three dimensions the table becomes harder to visualize and to render as a high-dimensional matrix. But an alternative approach that works for any number of discrete random variables is to construct a two-column table with one row per combination of possible values: the first column holds the combination (say, as a vector of values), the second column holds the corresponding probability of that combination.

In your hypothetical example we'd have one row for the combination $(3,H,7,A\heartsuit)$, another row for $(3,H,7,A\diamondsuit)$, another for $(3,H,8,A\heartsuit)$, and so on, for a total of $6\times2\times10\times4$ combinations. The result is a table with 480 rows.