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).
| Parameter | Range | Description |
|---|---|---|
| N₀ | ..., N₁ - 1, N₁ | Lower bound of outcomes |
| N₁ | N₀, N₀ + 1, ... | Upper bound of outcomes |
Probability Mass Function
Support
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)
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 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.