5.1.1 Two Discrete Random Variables

The probability mass function (pmf) of a single discrete rv $X$ specifies how much probability mass is placed on each possible $X$ value. The joint pmf of two discrete rv’s $X$ and $Y$ describes how much probability mass is placed on each possible pair of values $\left(x,y\right)$.

joint probability mass function

Let $X$ and $Y$ be two discrete rv’s defined on the sample space $\mathcal{S}$ of an experiment. The joint probability mass function $p\left(x,y\right)$ is defined for each pair of numbers $\left(x,y\right)$ by

$$p\left(x,y\right)=P\left(X=x\text{ and }Y=y\right)$$

It must be the case that $p\left(x,y\right)\ge 0$ and $\sum _x\sum _{y}p\left(x,y\right)=1$.

Now let $A$ be any particular set consisting of pairs of $\left(x,y\right)$ values

• e.g., $A=\{\left(x,y\right):x+y=5\}$ or $\{\left(x,y\right):\max \left(x,y\right)\le 3\}$ Then the probability $P\left[\left(X,Y\right)\in A\right]$ that the random pair $\left(X,Y\right)$ lies in the set $A$ is obtained by summing the joint pmf over pairs in $A$ :

$$P\left[\left(X,Y\right)\in A\right]=\sum _{\left(x,y\right)}\sum _{\in A}p\left(x,y\right)$$

EXAMPLE 5.1

Once the joint pmf of the two variables $X$ and $Y$ is available, it is in principle straightforward to obtain the distribution of just one of these variables.

Example

Let $X$ and $Y$ be the number of statistics and mathematics courses, respectively, currently being taken by a randomly selected statistics major. Suppose that we wish the distribution of $X$, and that when $X=2$, the only possible values of $Y$ are 0,1, and 2. Then

$$\begin{array}{rl}{p}_X(2)& =P(X=2)\\ & =P\left[(X,Y)=(2,0)\text{ or }(2,1)\text{ or }(2,2)\right]\\ & =p(2,0)+p(2,1)+p(2,2)\end{array}$$

That is, the joint pmf is summed over all pairs of the form $\left(2,y\right)$. More generally, for any possible value $x$ of $X$, the probability $p_X\left(x\right)$ results from holding $x$ fixed and summing the joint $\mathrm{pmf}p\left(x,y\right)$ over all $y$ for which the pair $\left(x,y\right)$ has positive probability mass. The same strategy applies to obtaining the distribution of $Y$ by itself.

marginal probability mass function

The marginal probability mass function of $X$, denoted by $p_X\left(x\right)$, is given by

$$p_X\left(x\right)=\sum _{y:p\left(x,y\right)>0}p\left(x,y\right)\text{ for each possible value }x$$

Similarly, the marginal probability mass function of $Y$ is

$$p_Y\left(y\right)=\sum _{x:p\left(x,y\right)>0}p\left(x,y\right)\text{ for each possible value }y.$$

The use of the word marginal here is a consequence of the fact that if the joint pmf is displayed in a rectangular table as in Example 5.1, then the row totals give the marginal pmf of $X$ and the column totals give the marginal pmf of $Y$. Once these marginal pmf’s are available, the probability of any event involving only $X$ or only $Y$ can be calculated.

EXAMPLE 5.2 (Example 5.1 continued)