Cumulative probabilities
The cumulative probability for any \(x\) is found by summing the probability function \(p(u)\) over all integer values \(u\) that are less than or equal to \(x\).
\[ F(x) = \sum_{u \le x} p(x) \]When considered as a function of \(x\), this is the distribution's cumulative distribution function.
Cumulative distribution function
The cumulative distribution function for the geometric distribution with probability function
\[ p(x) = \pi (1-\pi)^{x-1} \quad \quad \text{for } x = 1, 2, \dots \]is
\[ F(x) = \begin{cases} 1 - (1-\pi)^{\lfloor x \rfloor} & \text{for } x \ge 0 \\ 0 & \text{for } x \lt 0 \end{cases} \]where \(\lfloor x \rfloor\) denotes the largest integer less than or equal to \(x\).
For any integer \(x\), the probability that it will take more than \(x\) trials before the first success is observed is the probability that the first \(x\) trials are all failures,
\[ P(X \gt x) \;=\; (1-\pi)^x \]The cumulative distribution function is therefore
\[ F(x) \;=\; P(X \le x) \;=\; 1 - P(X \gt x) \;=\; 1 - (1-\pi)^x \]The result follows from noting that \(F(x)\) is the same for other values between integers \(x\) and \((x+1)\).
An alternative algebraic proof is shown below.
\[ \begin{align} F(x) & = p(0) + p(1) + \dots + p({\lfloor x \rfloor}) \\ & = 1 - \big( p(\lfloor x \rfloor + 1) + p(\lfloor x \rfloor + 2) + \dots \big) \\ & = 1 - \big( \pi(1-\pi)^{\lfloor x \rfloor} + \pi(1-\pi)^{\lfloor x \rfloor + 1} + \dots \big) \\ & = 1 - \pi(1-\pi)^{\lfloor x \rfloor}\big( 1 + (1-\pi) + (1-\pi)^2 + \dots \big) \end{align} \]Using the formula for the sum of this geometric series with \(a = (1 - \pi)\),
\[ \begin{align} F(x) & = 1 - \pi(1-\pi)^{\lfloor x \rfloor} \times \frac 1 {1 - (1-\pi)} \\ & = 1 - (1-\pi)^{\lfloor x \rfloor} \end{align} \]We now display the cumulative distribution function graphically.
Graphical display
The diagram below shows both the probability function of a geometric distribution, \(p(x)\), and its cumulative distribution function, \(F(x)\).
Click the bar for value 5 to highlight it and lower bars. The cumulative probability for this value is shown with a red line on the graph of the cumulative distribution function.
Observe that decreasing the probability of success, \(\pi\), also decreases the height of the cumulative distribution function — we become less likely to see the first success within the first 5 values.