Discrete Probability Distributions (Edexcel International A Level (IAL) Maths) : Revision Note

Dan Finlay

Last updated

Did this video help you?

Discrete Random Variables

What is a discrete random variable?

  • A random variable is a variable whose value depends on the outcome of a random event

    • The value of the random variable is not known until the event is carried out (this is what is meant by 'random' in this case)

  • Random variables are denoted using upper case letters (X , Y , etc )

  • Particular outcomes of the event are denoted using lower case letters ( x, y, etc)

  • begin mathsize 16px style straight P left parenthesis X equals x right parenthesis end style means "the probability of the random variable X taking the value x"

  • A discrete random variable (often abbreviated to DRV) can only take certain values within a set

    • Discrete random variables usually count something

    • Discrete random variables usually can only take a finite number of values but it is possible that it can take an infinite number of values (see the examples below)

  • Examples of discrete random variables include:

    • The number of times a coin lands on heads when flipped 20 times
      (this has a finite number of outcomes: 0,1,2,…,20)

    • The number of emails a manager receives within an hour
      (this has an infinite number of outcomes: 1,2,3,…)

    • The number of times a dice is rolled until it lands on a 6
      (this has an infinite number of outcomes: 1,2,3,…)

    • The number on a bingo ball when one is drawn at random
      (this has a finite number of outcomes: 1,2,3…,90)

Did this video help you?

Probability Distributions (Discrete)

What is a probability distribution?

  • A discrete probability distribution fully describes all the values that a discrete random variable can take along with their associated probabilities

    • This can be given in a table (similar to GCSE)

    • Or it can be given as a function (called a probability mass function)

    • They can be represented by vertical line graphs (the possible values for along the horizontal axis and the probability on the vertical axis)

  • The sum of the probabilities of all the values of a discrete random variable is 1

    • This is usually written begin mathsize 16px style ΣP left parenthesis X equals x right parenthesis equals 1 end style

  • A discrete uniform distribution is one where the random variable takes a finite number of values each with an equal probability

    • If there are n values then the probability of each one is begin mathsize 14px style 1 over n end style

4-1-1-discrete-probability-distributions-diagram-1

Did this video help you?

Cumulative Probabilities (Discrete)

How do I calculate probabilities using a discrete probability distribution?

  • For probability distributions that take a small number of values start by drawing a table to represent the probability distribution

    • If the distribution is given as a function then find each probability

    • If any probabilities are unknown then use algebra to represent them

      • Form an equation using sum straight P left parenthesis X equals x right parenthesis equals 1

  • To find P left parenthesis X equals k right parenthesis

    • If k is a possible value of the random variable X then P left parenthesis X equals k right parenthesis will be given in the table 

    • If kis not a possible value then begin mathsize 16px style P left parenthesis X equals k right parenthesis equals 0 end style

What is the cumulative distribution function?

  • The cumulative distribution function, denoted , is the probability that the random variable takes a value less than or equal to x.

    • straight F left parenthesis x right parenthesis equals straight P left parenthesis X less or equal than x right parenthesis

  • You may be asked to draw a table for the cumulative distribution function

    • This will be similar to a probability distribution function but instead the bottom row will be F(x)  instead of P(X = x)

How do I calculate cumulative probabilities?

  • To find begin mathsize 16px style P left parenthesis X less or equal than x right parenthesis end style (equivalently F(x))

    • Identify all possible values, begin mathsize 16px style x subscript i end style, that X can take which satisfy begin mathsize 16px style x subscript i less or equal than k end style

    • Add together all their corresponding probabilities

    • straight P left parenthesis X less or equal than k right parenthesis equals sum for x subscript i less or equal than k of straight P left parenthesis X italic equals x subscript i right parenthesis

  • Using a similar method you can find begin mathsize 16px style P left parenthesis X less than k right parenthesis comma space P left parenthesis X greater or equal than k right parenthesis space end styleand begin mathsize 16px style P left parenthesis X greater than k right parenthesis end style

  • As all the probabilities add up to 1 you can form the following equivalent equations:

    • begin mathsize 16px style straight P left parenthesis X less than k right parenthesis plus straight P left parenthesis X equals k right parenthesis plus straight P left parenthesis X greater than k right parenthesis equals 1 end style

  • To calculate more complicated probabilities such as begin mathsize 16px style straight P left parenthesis X squared less than 4 right parenthesis end style 

    • Identify which values of the random variable satisfy the inequality or event in the brackets

    • Add together the corresponding probabilities

How do I know which inequality to use?

  • begin mathsize 16px style straight P left parenthesis X less or equal than k right parenthesis end stylewould be used for phrases such as:

    • At most k, no greater than k, etc

  • would be used for phrases such as:

    • Fewer than k

  • would be used for phrases such as:

    • At least k  , no fewer than k, etc

  • would be used for phrases such as:

    • Greater than k, etc

Worked Example

The probability distribution of the discrete random variable X is given by the function

 P left parenthesis X equals x right parenthesis equals open curly brackets table row cell k x squared end cell cell x equals negative 3 comma negative 1 comma 2 comma 4 end cell row 0 otherwise end table close 

(a) Show that k equals 1 over 30.

(b) Calculate straight F left parenthesis 3 right parenthesis.

(c) Calculate straight P left parenthesis X squared space less than space 5 right parenthesis.

3-1-1-discrete-probability-distributions-we-solution-part-1
3-1-1-discrete-probability-distributions-we-solution-part-2
3-1-1-discrete-probability-distributions-we-solution-part-3

Examiner Tips and Tricks

  • Try to draw a table if there are a finite number of values that the discrete random variable can take

  • When finding a probability, it will sometimes be quicker to subtract the probabilities of the unwanted values from 1 rather than adding together the probabilities of the wanted values

  • Always make sure that the probabilities are between 0 and 1, and that they add up to 1!

👀 You've read 1 of your 5 free revision notes this week
An illustration of students holding their exam resultsUnlock more revision notes. It's free!

By signing up you agree to our Terms and Privacy Policy.

Already have an account? Log in

Did this page help you?

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

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.

Download notes on Discrete Probability Distributions