Definition of probability distribution of a DRV

A discrete random variable \(X\) is one with the following defining properties:

1. the set of values this variable can take is countable, i.e., either it is finite or it can be listed out in one-to-one correspondence with the positive integers, e.g., \(x_1, x_2, x_3, ..., x_n, ...\);
2. the associated probability of each point \(x_k\) is such that \(0 \le \mathrm{P}(X = x_k) \le 1\) for each \(k = 1, 2, ..., n, ...\); and
3. \[\sum\limits_{\mathrm{all} \, x} \mathrm{P}(X = x) = 1.\]

To construct a distribution, first define what the numerical variable counts or measures. List the possible values of that variable, then combine the probabilities of all outcomes producing each value. Equally likely elementary outcomes need not produce equally likely values of the random variable.

Check both conditions when finding unknown probabilities: all entries lie between zero and one, and their sum is one. An algebraic solution to the total-probability equation is invalid if it produces a negative entry. An event such as \(X>k\) is found by adding the entries for the admissible values.