Definition:Uniform Distribution/Discrete
Definition
Let $X$ be a discrete random variable on a probability space.
Then $X$ has a discrete uniform distribution with parameter $n$ if and only if:
- $\Img X = \set {1, 2, \ldots, n}$
- $\map \Pr {X = k} = \dfrac 1 n$
That is, there is a number of outcomes with an equal probability of occurrence.
This is written:
- $X \sim \DiscreteUniform n$
Also defined as
The discrete uniform distribution can also be defined as:
- A discrete random variable $X$ has a discrete uniform distribution with parameter $n + 1$ if and only if:
- $\Img X = \set {0, 1, \ldots, n}$
- $\map \Pr {X = k} = \dfrac 1 {n + 1}$
Examples
Coin Toss
Consider the exercise of coin-tossing.
There are $2$ possible outcomes: $\mathrm H$eads or $\mathrm T$ails, each with a probability of $\dfrac 1 2$.
Hence this is modelled by a discrete uniform distribution with parameter $2$.
Casting of Die
Consider the exercise of casting a fair die.
There are $6$ equally probable outcomes: $1$, $2$, $3$, $4$, $5$ and $6$, each with a probability of $\dfrac 1 6$.
Hence this is modelled by a discrete uniform distribution with parameter $6$.
Random Digits
Consider the exercise of selecting a random digit.
There are $10$ equally probable outcomes in the integer interval $\closedint 0 9$, each with a probability of $\dfrac 1 {10}$.
Hence this is modelled by a discrete uniform distribution with parameter $10$.
Also known as
A discrete uniform distribution is referred to in some sources as just a uniform distribution.
Such sources may also refer to a continuous uniform distribution as a rectangular distribution, and so meaning is not compromised.
Also see
- Results about the discrete uniform distribution can be found here.
Technical Note
The $\LaTeX$ code for \(\DiscreteUniform {n}\) is \DiscreteUniform {n} .
When the argument is a single character, it is usual to omit the braces:
\DiscreteUniform n
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): probability
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): uniform distribution: 1.
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): probability
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): uniform distribution: 1.
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): uniform distribution