Chapter 6 Some discrete distributions

All of the distributions we have considered so far are continuous, and therefore applicable for a continuous random variable. However, there are also some commonly used discrete distributions which are applicable to discrete random variables. They can help us model the types of experiments we considered earlier, and others. For example, the probability of receiving a positive return on a $100 stock market investment after one year, the probability of rolling six 'sixes' from 12 rolls of a die, or the the expected number of births overnight in a maternity hospital. For a discrete random variable \(X\), some common discrete distributions are represented below.

Distribution Parameters
\(X\sim \text{BERN}(p)\)
\(X\sim \text{BIN}(n, p)\)
\(n, p\)
Negative Binomial
\(X\sim \text{NB}(r, p)\)
\(r, p\)
\(X\sim \text{pois}(\lambda)\)

Rather than probability density functions, discrete distributions are expressed via probability mass functions (PMFs), since each value within the sample space has a probability 'mass' attached to it. Cumulative distribution functions and quantile functions can also be applied to discrete distributions. Since the random variable associated with these types of functions are discrete, it is important to be careful when calculating probabilities, since it is no longer true that \(P(X\leq x) = P(X < x)\).