Theoretical discrete distributions

The probability distribution $p(x)$ is completely defined by specifying the $k$ values $\{p(x_1),p(x_2),\ldots,p(x_k)\}$. However, in many cases, it is possible to obtain a very good approximation to the distribution by assuming a simple functional form $p(x)=f(x;\theta_1,\ldots,\theta_m)$ determined by a smaller number ($m<k$) of more meaningful population parameters $\{\theta_1,\theta_2,\ldots,\theta_m\}$. Over the years, statisticians have identified several theoretical probability distributions $p(x)=f(x;\theta_1,\ldots,\theta_m)$ that are very useful for modelling the probability distributions of observed data. These distributions are known as parametric probability models since they are completely determined by a few important parameters $\{\theta_1,\theta_2,\ldots,\theta_m\}$. The following subsections will briefly describe some of the functions that are used most frequently to model the probability distribution of discrete random variables.

Figure: Examples of discrete distributions: (a) Bernoulli $\pi=0.4$ , (b) Binomial with $n=15$ and $\pi=0.4$, and (c) Poisson with $\mu=6$.