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Discrete Cumulative Distribution Functions


We now learn eabout discrete cumulative probability distributions and cumulative distribution function.

At times, rather than having to calculate the probability of a specific value of \(X\) occurring, we'll need to calculate the probability that \(X\) be less than or equal to some value: \[P\begin{pmatrix}X \leq k \end{pmatrix}\] For such probabilities we'll need to use the cumulative distribution function.

Cumulative Distribution Function (CDF)


Given a discrete random variable \(X\), and its probability distribution function \(P \begin{pmatrix}X = x \end{pmatrix}=f(x)\), we define its cumulative distribution function, CDF, as: \[P \begin{pmatrix} X \leq k \end{pmatrix} = F(x)\] Where: \[F(x) = \sum_{t=x_{\text{min}}}^x P \begin{pmatrix} X = t \end{pmatrix}\] That's: \[F(x) = P \begin{pmatrix} X = x_{\text{min}} \end{pmatrix} + \dots + P \begin{pmatrix} X = x \end{pmatrix}\] Where \(x_{\text{min}}\) corresponds to the smallest value the discrete random variable \(X\) can take.

This function allows us to calculate the probability that the discrete random variable is less than or equal to some value \(x\).

We usually just write: \[P\begin{pmatrix} X \leq x \end{pmatrix} = \sum_{t=x_{\text{min}}}^x P \begin{pmatrix} X = t \end{pmatrix}\] Note: the fact that we are using a capital \(F\), is to highlight that fact that we are technically integrating (summing-up) \(f(t)\) from \(t=x_{\text{min}}\) to \(t=x\).

Don't panic! It's not as complicated as it looks!

Example

A wheel is numbered \(2\), \(3\), \(4\), \(5\) and \(6\), like the one illustrated in the sketch here.

An experiment consists of spinning the wheel and guessing the number on which the red pointer will point, when the wheel stops.

By defining the discrete random variable \(X\) as: "the number on which the red pointer will point, when the wheel stops", find the probability that the \(X\leq 4\).

In other words: find \(P\begin{pmatrix}X\leq 4\end{pmatrix}\).

Spin the Wheel! What number will it stop on?

Solution

To find \(P\begin{pmatrix} X \leq 4 \end{pmatrix}\) we use the cumulative distribution function and state: \[P\begin{pmatrix} X \leq 4 \end{pmatrix} = \sum_{x=2}^{4} P \begin{pmatrix} X = x \end{pmatrix}\] That's: \[P\begin{pmatrix} X \leq 4 \end{pmatrix} = P \begin{pmatrix} X = 2 \end{pmatrix} + P \begin{pmatrix} X = 3 \end{pmatrix} + P \begin{pmatrix} X = 4 \end{pmatrix}\] Finally we find: \[\begin{aligned} P\begin{pmatrix} X \leq 4 \end{pmatrix} & = \frac{2^2}{90} + \frac{3^2}{90} + \frac{4^2}{90} \\ & = \frac{4}{90} + \frac{9}{90} + \frac{16}{90} \\ & = \frac{29}{90} \\ P\begin{pmatrix} X \leq 4 \end{pmatrix} & = 0.32 \end{aligned}\] So \(P\begin{pmatrix} X \leq 4 \end{pmatrix} = 0.32\) (rounded to \(2\) decimal places).
This makes sense since the numbers \(2\), \(3\) and \(4\) cover just a little less than a third of the wheel, so the probability that the wheel stops on the area that they cover should also be just a little less than \(\frac{1}{3}\).

Tutorial 1

Exercise 1

  1. A discrete random variable \(X\) has probability distribution function defined by: \[P\begin{pmatrix} X = x \end{pmatrix} = \frac{x^2}{30} \] Where \(x = \left \{ 1, \ 2, \ 3, \ 4 \right \}\).
    Calculate the probability that \(X \leq 2\).

  2. A discrete randome variable \(X\) has probability distribution function defined by: \[f(x) = \frac{x}{15}\] Where \(x = \left \{ 1, \ 2, \ 3, \ 4, \ 5 \right \}\).
    Calculate the probability that \(X < 4 \).

Answers Without Working

  1. \(P \begin{pmatrix} X \leq 2 \end{pmatrix} = \frac{1}{6}\)

  2. \(P \begin{pmatrix} X < 4 \end{pmatrix} = 0.4 \)

Using the Complement

We now know how to calculate the probability that \(X\) be less than, or less than or equal to, a certain value.
But what if we were asked to find the probability that \(X\) be greater than a certain value, or the probability that \(X\) is "at least" some amount?

To answer such questions we'll need to use the complement formula, which we show here:

Complement Formula

To calculate the probability \(P\begin{pmatrix} X > a \end{pmatrix}\) we use the complement formula, which states: \[P\begin{pmatrix} X > a \end{pmatrix} = 1 - P\begin{pmatrix} X \leq a \end{pmatrix}\]

Example

A discrete random variable, \(X\), has a probability distribution function defined as: \[f(x) = P \begin{pmatrix} X = x\end{pmatrix} = \frac{x^3}{36}\] Find the probability that \(X > 0 \).

Illustrating Cumulative Distributions

Now that we know how to use cumulative distribution formula, we learn how to illustrate them.
This can be done in either:

  • a cumulative distribution table, or
  • a cumulative distribution bar chart.
We learn about each of these here.

Cumulative Distribution Tables and Charts

Further Techniques

At times we'll need to calculate the probability that the discrete random variable is between two specfic values, a lower bound and an upper bound.
Typically we'll need to calculate: \[P \begin{pmatrix} a < X \leq b \end{pmatrix} \] To do this we'll need the formula we learn here:

\[\begin{aligned} P \begin{pmatrix} a < X \leq b \end{pmatrix} &= F(b) - F(a) \\ & = \sum_{x = x_{\text{min}}}^b f(x) - \sum_{x = x_{\text{min}}}^a f(x) \\ P \begin{pmatrix} a < X \leq b \end{pmatrix} &= \sum_{x = x_{\text{min}}}^b P \begin{pmatrix} X = x \end{pmatrix} - \sum_{x = x_{\text{min}}}^a P\begin{pmatrix} X = x \end{pmatrix}\ \end{aligned}\]