Editing Random variable (section)

===Discrete random variable===

Consider an experiment where a person is chosen at random. An example of a random variable may be the person's height.  Mathematically, the random variable is interpreted as a function which maps the person to their height.  Associated with the random variable is a probability distribution that allows the computation of the probability that the height is in any subset of possible values, such as the probability that the height is between 180 and 190&nbsp;cm, or the probability that the height is either less than 150 or more than 200&nbsp;cm.

Another random variable may be the person's number of children; this is a discrete random variable with  non-negative integer values.  It allows the computation of probabilities for individual integer values – the probability mass function (PMF) – or for sets of values, including infinite sets.  For example, the event of interest may be "an even number of children".  For both finite and infinite event sets, their probabilities can be found by adding up the PMFs of the elements; that is, the probability of an even number of children is the infinite sum <math>\operatorname{PMF}(0) + \operatorname{PMF}(2) + \operatorname{PMF}(4) + \cdots</math>.

In examples such as these, the [[sample space]] is often suppressed, since it is mathematically hard to describe, and the possible values of the random variables are then treated as a sample space.  But when two random variables are measured on the same sample space of outcomes, such as the height and number of children being computed on the same random persons, it is easier to track their relationship if it is acknowledged that both height and number of children come from the same random person, for example so that questions of whether such random variables are correlated or not can be posed.

If <math display = "inline">\{a_n\}, \{b_n\}</math> are countable sets of real numbers, <math display="inline">b_n >0</math> and <math display="inline">\sum_n b_n=1</math>, then <math display="inline"> F=\sum_n b_n \delta_{a_n}(x)</math> is a discrete distribution function. Here <math>  \delta_t(x) = 0</math> for <math> x < t</math>, <math> \delta_t(x) = 1</math> for <math> x \ge t</math>. Taking for instance an enumeration of all rational numbers as <math>\{a_n\}</math> , one gets a discrete function that is not  necessarily  a [[step function]] ([[piecewise]] constant).
====Coin toss====

The possible outcomes for one coin toss can be described by the sample space <math>\Omega = \{\text{heads}, \text{tails}\}</math>. We can introduce a real-valued random variable <math>Y</math> that models a $1 payoff for a successful bet on heads as follows:
<math display="block">Y(\omega) = \begin{cases}
1, & \text{if } \omega = \text{heads}, \\[6pt]
0, & \text{if } \omega = \text{tails}.
\end{cases}</math>

If the coin is a [[fair coin]], ''Y'' has a [[probability mass function]] <math>f_Y</math> given by:
<math display="block">f_Y(y) = \begin{cases}
\tfrac 12,& \text{if }y=1,\\[6pt]
\tfrac 12,& \text{if }y=0,
\end{cases}</math>

====Dice roll====

[[File:Dice Distribution (bar).svg| right | thumb | If the sample space is the set of possible numbers rolled on two dice, and the random variable of interest is the sum ''S'' of the numbers on the two dice, then ''S'' is a discrete random variable whose distribution is described by the [[probability mass function]] plotted as the height of picture columns here.]]

A random variable can also be used to describe the process of rolling dice and the possible outcomes. The most obvious representation for the two-dice case is to take the set of pairs of numbers ''n''<sub>1</sub> and ''n''<sub>2</sub> from {1, 2, 3, 4, 5, 6} (representing the numbers on the two dice) as the sample space. The total number rolled (the sum of the numbers in each pair) is then a random variable ''X'' given by the function that maps the pair to the sum:
<math display="block">X((n_1, n_2)) = n_1 + n_2</math>
and (if the dice are [[fair die|fair]]) has a probability mass function ''f''<sub>''X''</sub> given by:
<math display="block">f_X(S) =  \frac{\min(S-1, 13-S)}{36}, \text{ for } S \in \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12\}</math>