<Li> A random function F (\ displaystyle F) may be represented as a collection of random variables F (x) (\ displaystyle F (x)), giving the function's values at the various points x (\ displaystyle x) in the function's domain . The F (x) (\ displaystyle F (x)) are ordinary real - valued random variables provided that the function is real - valued . For example, a stochastic process is a random function of time, a random vector is a random function of some index set such as 1, 2,..., n (\ displaystyle 1, 2, \ ldots, n), and random field is a random function on any set (typically time, space, or a discrete set). </Li> <P> In an experiment a person may be chosen at random, and one random variable may be the person's height . Mathematically, the random variable is interpreted as a function which maps the person to the person's 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 cm, or the probability that the height is either less than 150 or more than 200 cm . </P> <P> 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 PMF ⁡ (0) + PMF ⁡ (2) + PMF ⁡ (4) + ⋯ (\ displaystyle \ operatorname (PMF) (0) + \ operatorname (PMF) (2) + \ operatorname (PMF) (4) + \ cdots). </P> <P> In examples such as these, the sample space (the set of all possible persons) 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 . </P>

State the two conditions for a valid discrete probability distribution