Discrete Uniform Distribution (Edexcel International A Level Maths): Revision Note

Discrete Uniform Distribution

What is a discrete uniform distribution?

• A discrete uniform distribution is a discrete probability distribution

• The discrete random variable X follows a discrete uniform distribution if

  • There are a finite number of distinct outcomes (n)

  • Each outcome is equally likely

• If there are n distinct outcomes,  

• In many cases the outcomes of X are the integers 1, 2, 3, .., n

  • for 

  • 0 for any other value of X

• The distribution can be represented visually using a vertical line graph where the lines have equal heights

What is the mean and variance of a discrete uniform distribution?

• If the outcomes of X are the integers 1, 2, 3, …, n

  • The expected value (mean) is 

  • The variance is 

    • Square root to get the standard deviation

• The discrete uniform distribution is symmetrical so the median is the same as the mean

  • There is no mode as each value is equally likely

Do the outcomes have to be 1 to n?

• The numbers can be anything as long as they are equally likely

• The formulae for the mean and variance only apply when the values are the integers 1 to n

• If the outcomes form an arithmetic sequence then the distribution can be transformed to the distribution with the values 1 to n

• If X is the discrete uniform distribution using 1 to n and Y is a discrete uniform distribution whose outcomes form an arithmetic sequence then:

  • Y = aX + b

• You can then use this formula to find the mean and variance

  • E(Y) = aE(X) + b

  • Var(Y) = a² Var(X)

• For example: Y = 2, 5, 8, 11 can be transformed to X = 1, 2, 3, 4 using Y = 3X - 1

What can be modelled using a discrete uniform distribution?

• Anything which satisfies the two conditions

  • finite distinct outcomes and all equally likely

• For example, let R be the second digit of a number given by a random number generator

  • There are 10 distinct outcomes: 0, 1, 2, ..., 9

  • As it is a random number then each value is equally likely to be the second digit

What can not be modelled using a discrete uniform distribution?

• Anything where the number of outcomes is infinite

  • The number obtained when a person is asked to write down any integer

• Anything where the outcomes are not equally likely

  • The number obtained when one of the first 5 Fibonacci numbers is randomly selected

    • 1, 1, 2, 3, 5

    • 1 appears twice so is more likely to be picked than the rest