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

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)

• means "the probability of the random variable X taking the value "

• 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)

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 

• 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 

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 

• To find 

  • If k is a possible value of the random variable X then  will be given in the table 

  • If is not a possible value then 

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.

  • 

• 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  (equivalently F(x))

  • Identify all possible values, , that X can take which satisfy 

  • Add together all their corresponding probabilities

  • 

• Using a similar method you can find and 

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

  • 

  • 

  • 

• To calculate more complicated probabilities such as  

  • 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?

• would 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