Definition

The probability distribution of a discrete variable $X$ is the set of probabilities $p(x)=\Pr(X=x)$ for all the possible values $x$ of $X$ in the event space. So for a discrete random variable that can take $k$ distinct values in the set $\{x_1, x_2, \ldots, x_k\}$, the probability distribution is defined by the $k$ probabilities $\{\Pr(X=x_1),\Pr(X=x_2),\ldots,\Pr(X=x_k)\}$. The probability distribution contains complete information about ALL the statistical properties of $X$; for example, once the probability distribution is known, the expectation of any function of $X$ can be calculated.