2.2 Probability functions for discrete random variables

For discrete random variables, the probability function is called a probability mass function (PMF).

Example 2.4 (Discrete uniform distribution) A rv $X$ has the uniform discrete distribution defined from $a$ to $b$ if the PMF is $f_X(x) = \left\{ \begin{array}{ll} \displaystyle \frac{1}{b-a+1} & \text{if $x = a, a+1, \dots, b$};\\[6pt] 0 & \text{otherwise}. \end{array} \right.$ This is usually written, for convenience, as $f_X(x) = \displaystyle \frac{1}{b-a+1}\quad \text{for $x=a, a+1, \dots, b$}.$

A , say $f(x)$ , has two properties:

$\displaystyle \sum_{-\infty}^{\infty} f_X(x) = 1$ ; that is, the total probability is one (in other words, $x$ certainly takes some value).
$f_X(x) \ge 0$ for all $x$ ; that is, probabilities are non-negative.

Probabilities are found by summing over appropriate values.

Example 2.5 (Discrete uniform distribution) A rv $X$ has the uniform discrete distribution defined from $3$ to $7$ inclusive if the PMF is $f_X(x) = \displaystyle \frac{1}{7-3+1} = \frac{1}{5} \quad\text{for $x=3, 4, 5, 6, 7$}.$ The probability that the rv $X$ takes a value between $5$ and $7$ inclusive is $\Pr(5\le X\le 7) = \sum_{5}^{7} \frac{1}{5} = 0.6.$

Example 2.6 (Discrete PMF) Consider a rv $Z$ with PMF $f_Z(z) = \frac{|z - 3|}{4} \quad\text{for $z=1, 2, 3, 4$}.$ The probability that $Z>2$ is $\Pr(Z>2) = \Pr(Z=3) + \Pr(Z=4) = \frac{|3-3|}{4} + \frac{|4-3|}{4} = 0.25;$ see Fig. 2.3.

FIGURE 2.3: The discrete distribution for $Z$