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. \[\sum_{\mathrm{all} \, x} \mathrm{P}(X = x) = 1.\]