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