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The discrete uniform distribution gives each of the integer values N₀, N₀ + 1, ..., N₁ the same probability of success. Since the probabilities of all the outcomes must add up to 1, the probability of each outcome is 1/(N₁ − N₀ + 1).

The function ⌊x⌋ in the cdf denotes the "floor" or greatest integer function.

Parameter Range Description
N₀ ..., N₁ − 1, N₁ Lower bound of outcomes
N₁ N₀, N₀ + 1, ... Upper bound of outcomes

Probability Mass Function

P ( X = x ; N 0 , N 1 ) = 1 N 1 − N 0 + 1
F ( x ; N 0 , N 1 ) = ⌊ x ⌋ − N 0 + 1 N 1 − N 0 + 1

Support

x = N 0 , … , N 1

Mean

Variance

Example N₀ N₁
A fair six-sided die is thrown. Let X be the number of dots showing. 1 6
A card is chosen at random from a standard 52 card deck. Let X = 1 if a heart is chosen, 2 if a club, 3 if a diamond, and 4 if a spade. 1 4
In the Powerball lottery, there are 26 red Powerballs numbered from 1 to 26. Let X be the number of the red Powerball in a single draw. 1 26

X ∼ Discrete Uniform(N₀, N₁)

N₀ = N₁ =
Chart of the discrete uniform distribution Chart area for displaying the discrete uniform pmf, cdf, visualization, and simulation

E(X) = , Var(X) =

The discrete uniform distribution arises naturally when we have a finite number of possible outcomes and no reason to assume that any one outcome is more likely than any other.

The discrete uniform distribution arises naturally when we have a finite number of possible outcomes and no reason to assume that any one outcome is more likely than any other.

The illustration above shows an integer X chosen randomly from the set {N₀, …, N₁}, where each integer has the same probability of being chosen.

The simulation above shows a number U chosen randomly from the interval [N₀ − 0.5, N₁ + 0.5]. The light blue line shows U rounded to the nearest integer, which has a discrete uniform(N₀, N₁) distribution. The histogram accumulates the results of each simulation.

Y = Discrete Uniform(0, 1) ∼ Bernoulli(0.5)

Chart of the related distribution Chart area for displaying the related pdf, cdf, and simulation

E(Y) = , Var(Y) =

The discrete uniform(N₀, N₁) distribution is shown in red superimposed over the Bernoulli(0.5). It can be seen that these are equal when N₀ = 0 and N₁ = 1. The reason is that we then have only two choices, each with probability 0.5. If the first choice is a "failure" and the second choice a "success", we have the definition of a Bernoulli(0.5) random variable.