Uniform Distribution

The uniform distribution, also known as the discrete uniform distribution, models a situation where a fixed number of outcomes all have an equal probability of occurring. It is used to represent scenarios where each outcome is equally likely.

The uniform distribution has two parameters: the lower limit $a$ and the upper limit $b$. The upper limit is exclusive, which means the highest possible value is actually one less than the upper limit. The probability mass function (PMF) is given by:

$$P(X=k)=\frac{1}{b-a}$$

The cumulative distribution function (CDF) is:

$$F(k)=\frac{k-a+1}{b-a}$$

The domain of the uniform distribution is $k\in \{a,a+1,\dots ,b-1\}$. The parameters must satisfy $a<b$.

The uniform distribution is widely used in various fields due to its simplicity and fundamental nature. Common applications include:

• Coin tossing when the coin is known to be unbiased has a uniform distribution with two possible values: 0 ('heads') and 1 ('tails'). The lower limit is 0. The upper limit is 2.

• When rolling dice, the score has a uniform distribution with lower limit 1 and upper limit 7.

• When choosing an element at random from a collection of $n$ elements, the (zero-based) index has a uniform distribution with lower limit 0 and upper limit $n$.

The uniform distribution has several important statistical properties:

The uniform distribution is closely related to several other distributions:

• The Binomial Distribution can be approximated by the uniform distribution when the number of trials is very large and the probability of success is very small.

• The Exponential Distribution models the time between successive occurrences of events that follow a Poisson process.

The discrete uniform distribution is implemented by the DiscreteUniformDistribution class. It has two constructors. The first constructor takes one argument: the upper limit of the distribution. This limit is exclusive. A sample from the distribution is always strictly smaller than the upper limit. The following constructs a discrete uniform distribution with samples in the range 0 to 4, inclusive:

var uniform1 = new DiscreteUniformDistribution(5);

Dim uniform1 = New DiscreteUniformDistribution(5)

No code example is currently available or this language may not be supported.

let uniform1 = DiscreteUniformDistribution(5)

The second constructor takes two arguments. The first argument is the lower limit for the distribution. The second parameter is the upper limit of the distribution. The following constructs a discrete uniform distribution for the number of eyes when rolling one die:

var uniform2 = new DiscreteUniformDistribution(1, 7);

Dim uniform2 = New DiscreteUniformDistribution(1, 7)

No code example is currently available or this language may not be supported.

let uniform2 = DiscreteUniformDistribution(1, 7)

The DiscreteUniformDistribution class has two specific properties: LowerBound and UpperBound, which return the lower and upper limits of the distribution.

DiscreteUniformDistribution has one static (Shared in Visual Basic) method, Sample, which generates a random sample using a user-supplied uniform random number generator. It has two overloads, corresponding to each of the two constructors.

var random = new Pcg32();
int sample1 = DiscreteUniformDistribution.Sample(random, 4);
int sample2 = DiscreteUniformDistribution.Sample(random, 1, 7);

Dim random = New Pcg32()
Dim sample1 = DiscreteUniformDistribution.Sample(random, 4)
Dim sample2 = DiscreteUniformDistribution.Sample(random, 1, 7)

No code example is currently available or this language may not be supported.

let random = Pcg32()
let sample1 = DiscreteUniformDistribution.Sample(random, 4)
let sample2 = DiscreteUniformDistribution.Sample(random, 1, 7)

The above example uses the Pcg32 class to generate uniform random numbers.

For details of the properties and methods common to all discrete probability distribution classes, see the topic on Discrete Probability Distributions.

• "Introduction to Probability Models" by Sheldon M. Ross.

• "Probability and Statistics" by Morris H. DeGroot and Mark J. Schervish.

Other Resources