The sum of the probabilities of all x values in a discrete distribution equals

Author: Daniel

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.

Video Transcript

Let 𝑋 denote a discrete random variable which can take the values one, two, three, four, five, and six. Given that 𝑋 has probability distribution function 𝑓 of 𝑥 equals 𝑎𝑥 over five, find the value of 𝑎.

In order to answer this problem, let’s recall some of the key properties of discrete random variables. When described by a probability distribution function 𝑓 of 𝑥, we know that the sum of all 𝑓 of 𝑥 values must be equal to one. In other words, the sum of all possible probabilities is equal to one. We also know that each individual value must be in the closed interval from zero to one. In other words, it can be no smaller than zero and no greater than one. So this helps us answer the problem. We are going to begin by finding the corresponding probability of each event occurring, in other words, the probability that 𝑋 takes any of the values given. Then, we’ll have expressions in terms of 𝑎 which we can sum to one and solve for 𝑎.

We’ll begin by constructing a table showing all the possible values of 𝑓 of 𝑥. 𝑓 of 𝑥 is given by the expression 𝑎𝑥 over five. So when 𝑋 is equal to one, we get 𝑎 times one over five, which is simply 𝑎 over five. Then when 𝑋 is equal to two, it’s 𝑎 times two over five, which is two 𝑎 over five. And then, in a similar way, it’s three 𝑎 over five when 𝑋 is equal to three, four 𝑎 over five, five 𝑎 over five, and six 𝑎 over five. Now we could simplify the expression five 𝑎 over five, but we’re actually going to add all of these values. We know they sum to one, so 𝑎 over five plus two 𝑎 over five plus three 𝑎 over five plus four 𝑎 over five plus five 𝑎 over five plus six 𝑎 over five equals one. These sum to 21𝑎 over five, so 21𝑎 over five equals one.

To find the value of 𝑎, we could divide by 21 over five. Or equivalently, we can do this in two steps by multiplying by five and then dividing by 21 to give us 𝑎 equals five over 21. It is worth double-checking that this value of 𝑎 satisfies the second property; that is, any individual probability must be greater than or equal to zero and less than or equal to one. Beginning with 𝑎 over five, when 𝑎 is five over 21, 𝑎 over five is one over 21. And this is in the correct interval. Two 𝑎 over five is two over 21, three 𝑎 over five is three over 21, four 𝑎 over five is four over 21, and so on. Every single value of 𝑓 of 𝑥 is between zero and one inclusive. And in fact, they sum to make one as required. 𝑎 is equal to five over 21.

In this section we learn about discrete random variables and probability distribution functions, which allow us to calculate the probabilities associated to a discrete random variable.

We start by defining discrete random variables and then define their probability distribution functions (pdf) and learn how they are used to calculate probabilities.

Tutorial

In the following tutorial, we learn more about what discrete random variables and probability distribution functions are and how to use them. Watch it before carrying-on.

Definition: Discrete Random Variable

Discrete Variables

A discrete variable is a variable that can "only" take-on certain numbers on the number line.
We usually refer to discrete variables with capital letters: \[X, \ Y, \ Z, \ \dots \] A typical example would be a variable that can only be an integer, or a variable that can only by a positive whole number.

Discrete variables can either take-on an infinite number of values or they can be limited to a finite number of values.

For instance the number we obtain , when rolling a dice is a discrete variable, which is limited to a finite number of values:\(1, \ 2, \ 3, \ 4, \ 5, \) or \(6\).

An example of a discrete variable that can take-on an "infinite" number of values could be: the number of rain drops that fall over a square kilometer in Sweden on November 25th.
Note: although this quantity can technically not be infinite, it is common practice and acceptable to assume so.

Discrete Random Variables

A discrete variable is a discrete random variable if the sum of the probabilities of each of its possible values is equal to \(1\).

Example

When we roll a single dice, the possible outcomes are: \[1, \ 2, \ 3, \ 4, \ 5, \ 6\] The probability of each of these outcomes is \(\frac{1}{6}\).
If we define the discrete variable \(X\) as: \(X:\) the number obtained when rolling a dice. Then this is a discrete random variable since the sum of the probabilities of each of these possible outcomes is equal to \(1\), indeed: \[\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6}+\frac{1}{6} = 1 \]

Probability Distribution Function (PDF)

Given a discrete random variable, \(X\), its probability distribution function, \(f(x)\), is a function that allows us to calculate the probability that \(X=x\).
In other words, \(f(x)\) is a probability calculator with which we can calculate the probability of each possible outcome (value) of \(X\). \[P\begin{pmatrix}X = x \end{pmatrix} = f(x) \]

Example

A bag contains several balls numbered either: \(2\), \(4\) or \(6\) with only one number on each ball. A simple experiment consists of picking a ball, at random, out of the bag and looking at the number written on the ball.

Defining the discrete random variable \(X\) as: \(X\): the number obtained when we pick a ball at random from the bag and given that its probability distribution function is: \[P\begin{pmatrix} X = x \end{pmatrix} = \frac{8x-x^2}{40}\] Answer each of the following:

1. State the possible values that \(X\) can take.
2. Calculate the probability of picking a ball with \(2\) on it.
3. Calculate the probability of picking a ball with \(4\) on it.

Distribution Tables & Graphs

To illustrate the probabilities of each of the possible values a discrete random variable \(X\) can take, it will often be useful to showcase all the possible values of \(X\) alongside the corresponding probability.
This is usually done in either:

• a probability distribution table, or
• a bar chart.

Each of these is illustrated in the following tutorial and in the detailed example below.

Example: Distribution Table & Graph

We'll stick to the example we saw further up:

A game of chance consists of picking, at random, a ball from a bag. Each ball is numbered either \(2\), \(4\) or \(6\). The discrete random variable is defined as: \(X\): the number obtained when we pick a ball from the bag.
The probability distribution function associated to the discrete random variable is: \[P\begin{pmatrix} X = x \end{pmatrix} = \frac{8x-x^2}{40}\]

1. Construct a probability distribution table to illustrate this distribution.
2. Draw a bar chart to illustrate this probability distribution.
3. Use the distribution table and bar chart to determine which value the discrete random variable \(X\) is most likely to take.

Tutorial

In the following tutorial we learn how to construct probability distributions tables and their corresponding bar charts. Make sure to watch before working through the exercises below.

Exercise

A discrete random variable \(X\) can take either of the values: \[x = \left \{ 2, \ 4, \ 6 \right \}\] and has a probability distribution function (pdf) defined as: \[P\begin{pmatrix} X = x \end{pmatrix} = \frac{8x-x^2}{40}\]

1. Construct a probability distribution table for \(X\).
2. Illustrate this probability distribution with a bar chart.
3. Using your previous answers, state which value the discrete random variable \(X\) most likely to take?

1. A discrete random variable \(X\) has a probability distribution function defined as: \[P\begin{pmatrix} X = x \end{pmatrix} = kx^2\] where: \(x = \left \{ 0, \ 1, \ 2, \ 3\right \}\).
   1. Find the value of \(k\).
   2. Calculate the probability that \(X = 2\).


2. A discrete random variable \(X\) has a probability distribution function defined as: \[P \begin{pmatrix} X = x \end{pmatrix} = \frac{x}{k} \] where: \(x = \left \{ 1, \ 2, \ 3, \ 4, \ 5 \right \}\).
   1. Find the value of \(k\).
   2. Illustrate this discrete probability distribution in a table.


3. A discrete random variable has a probability distribution function \(f(x)\), its distribution is shown in the following table:
   1. Find the value of \(k\) and draw the corresponding distribution table.
   2. Represent this distribution in a bar chart.
   3. Which value is the discrete random variable most likely to take?

Answers Without Working

1. 1. \(k = \frac{1}{14}\)
   2. \(P \begin{pmatrix} X = 2 \end{pmatrix} = \frac{2}{7}\) that's \(0.286\) (rounded to 3 significant figures).


2. 1. \(k = 15\)
3. 
   
   1. \(k=-0.1\)

      The probability distribution therefore becomes:
   2. The graphical representation, of this distribution, is shown in the following bar chart:

   3. The discrete random variable is most likely to take the value \(2\).

What is the sum of all probabilities in a discrete probability distribution?

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