Discrete Uniform Distribution (Edexcel International A Level Maths: Statistics 1)

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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,  P left parenthesis X equals x right parenthesis equals 1 over n
  • In many cases the outcomes of X are the integers 1, 2, 3, .., n
    • P left parenthesis X equals x right parenthesis equals 1 over n for begin mathsize 16px style n equals 1 comma space 2 comma space 3 comma space... comma n end style
    • 0 for any other value of X
  • The distribution can be represented visually using a vertical line graph where the lines have equal heights

3-1-4-discrete-uniform-diagram-1

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 begin mathsize 16px style fraction numerator n plus 1 over denominator 2 end fraction end style
    • The variance is fraction numerator size 16px n to the power of size 16px 2 size 16px minus size 16px 1 over denominator size 16px 12 end fraction
      • 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

Worked example

Each odd number from 1 to 99 is written on an individual tile and one is chosen at random. The random variable T represents the number on the chosen tile.

(a)       Find E left parenthesis T right parenthesis.

(b)       Find Var left parenthesis T right parenthesis.

(a)       Find E left parenthesis T right parenthesis.

 3-1-4-discrete-uniform-we-solution-part-1

3-1-4-discrete-uniform-we-solution-part-2

(b)       Find Var left parenthesis T right parenthesis.

3-1-4-discrete-uniform-we-solution-part-3

Exam Tip

  • Always check your mean and variance makes sense. If the numbers go from 1 to 100 then a mean of 101 is not possible!

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Dan

Author: Dan

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.

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