3.1.4 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

Worked Example

(a)       Find .

 

(b)       Find .

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