2.3 Discrete Random Variables

Figure 2.1 (Link) shows the CDF for a particular random variable, $R$ (a Poisson random variable with $\lambda =2$). It is horizontal, except at $r\in \{0,1,2,3,4,\mathrm{\dots },\}$, at which point it jumps (is discontinuous).

Random variables with CDFs which are horizontal except at jump points are called discrete random variables.

The probability of outcome $r$ ($r$ an integer) for a discrete random variable $R$ is given by the probability mass function (pmf) defined by

The probability $𝖯\left(R=r\right)$ is, therefore, only non-zero where there are jumps. The cdf in Figure 2.1 (Link) corresponds to non-zero probabilities for $r\in \{0,1,2,3,4,\mathrm{\dots },\}$. For $r\notin \{0,1,2,3,4,\mathrm{\dots },\}$ we have $𝖯\left(R=r\right)=0$.

A discrete random variable $R$ has a countable sample space (set of possible values), often only the integers or the non-negative integers. For simplicity we will take the sample space to be the integers in the following presentation.

Discrete random variables arise in a variety of ways:

1. 1.

   from experiments with a natural integer valued outcome (e.g. rolling a die),

2. 2.

   from experiments with outcomes to which integer values are assigned.

To write the cdf in terms of the pmf as a function of $r$ for any $r$ we need the function $\mathrm{int}(r)$, which denotes the largest integer smaller than or equal to $r$, e.g. $\mathrm{int}(3.9)=3$, $\mathrm{int}(2)=2$, $\mathrm{int}(-1.5)=-2$.

If $p(r)$ is a probability mass function then

1. 1.

   $0\le p(r)\le 1$ for all $r$,

2. 2.

   ${\sum }_{r=-\mathrm{\infty }}^{\mathrm{\infty }}p(r)=1$.

Solution. 

1. (a)

   $p(5)=-1/5$;

2. (b)

   $p(4)=2$;

3. (c)

   $p(1)+p(2)+p(3)+p(4)=1/2$.

For any event $A$ defined as a set of possible values of the random variable $R$, then

E.g. if $A=\{i:0\le i\le m\}$ then $𝖯\left(A\right)={\sum }_{i=0}^{m}p(i)$.