Mathematics and Statistics · Ch 16 — Probability Distributions

Probability Distribution of a Discrete Random Variable (p.m.f.)

For a discrete random variable $X$ taking the values $x_1, x_2, \ldots, x_n$ (or countably many values), the probability mass function (p.m.f.) states the probability of each value:

P(X = x_i) = p_i, \qquad i = 1, 2, \ldots, n.

Listing every value alongside its probability gives the probability distribution of $X$ , usually shown as a table:

$X = x$	$x_1$	$x_2$	$\cdots$	$x_n$
$P(X=x)$	$p_1$	$p_2$	$\cdots$	$p_n$

Note

Conditions for a valid p.m.f.

A function $p(x) = P(X=x)$ is a legitimate probability mass function iff both hold:

$p_i \ge 0$ for every $i$ (no probability is negative), and
$\displaystyle\sum_{i} p_i = 1$ (the probabilities of all possible values add up to exactly $1$ ).

These two conditions are the workhorse of nearly every discrete problem: they let us find an unknown constant in a p.m.f. (impose $\sum p_i = 1$ ) and they let us check our own work (the column of probabilities must total $1$ ).

Once the distribution is known, the probability of any event about $X$ is found by adding the probabilities of the values in that event. For instance,

$P(X \le 2) = P(X=0) + P(X=1) + P(X=2), \qquad P(a \le X \le b) = \sum_{a \le x_i \le b} p_i.$ …

Definition 4Probability mass function (p.m.f.)

For a discrete $X$ , the function $p(x)=P(X=x)$ giving the probability of each value; valid iff $p(x)\ge 0$ for all $ …

Definition 5Probability distribution

The complete listing (usually a table) of every value of a discrete random variable together wit …

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