A discrete random variable's probability function gives probabilities for all of its possible values. Probabilities for more complex events can be found by summing it over the relevant values. A similar quantity describes the joint distribution of two discrete random variables.
Definition
For two discrete random variables \(X\) and \(Y\), the joint probability function gives the probabilities for all possible combinations of values of the two variables,
\[ p(x, y) \;=\; P(X=x \textbf{ and } Y=y) \]A joint probability function can often be expressed as a single mathematical formula, but a 2-dimensional table of probabilities is an alternative.
Question
Consider a weighted six-sided die for which the value "6" has twice the probability of the other values. If the die is rolled twice, with \(X\) and \(Y\) being the values that appear on the first and second rolls, what is the joint probability function of the two variables?
(Solved in full version)
Joint probability functions must satisfy two properties:
Properties of joint probability functions
\[ p(x,y) \ge 0 \text{ for all } x,y \] \[ \sum_{\text{all } x,y} p(x,y) = 1 \](Proved in full version)
Probabilities for events about \(X\) and \(Y\)
The probabilities of other events can be found by adding joint probabilities. For any event, \(A\),
\[ P(A) \;\;=\;\; \sum_{(x,y) \in A} {p(x,y)} \]Question
In the above weighted dice example, what is the probability that the sum of the two dice will be ten or more?
(Solved in full version)