Discrete Random Variables - Cumulative Distribution Function

The cumulative distribution function of a random variable \(X\) is a function \(F_X\) that, when evaluated at a point \(x\), gives the probability that the random variable will take on a value less than or equal to \(x:\) \(\text{Pr}[X \leq x]\). For example, a random variable representing a single dice roll has cumulative distribution function

\[\text{Pr}[X \leq x] = F_X(x) = \begin{cases} \frac{1}{6} & x = 1 \\ \frac{2}{6} & x = 2 \\ \frac{3}{6} & x = 3 \\ \frac{4}{6} & x = 4 \\ \frac{5}{6} & x = 5 \\ \frac{6}{6} & x = 6. \end{cases} \]

For a discrete random variable, the cumulative distribution is defined by

\[F(x) = P(X \leq x) = \sum_{x_i \leq x}P(X = x_i) = \sum_{x_i \leq x}p(x_i),\]

where \(p\) is the p.d.f. of \(X\). This distribution is not continuous, and is constant between the \(x_i\).

Furthermore, the p.d.f. is related to the cumulative distribution by

\[\text{Pr}[X=x] = F_X(x) - F_X(x'),\]

where \(x'\) is the next smallest possible value of \(x\). In the case of a random variable defined on integers (as is typical), \(x'=x-1\). This forms the intuition for the relationship between the continuous p.d.f. and continuous cumulative distribution, where the p.d.f. is the derivative of the c.d.f.

• Random Variables