Discrete Uniform Distribution

The Probabilistic Experiment Behind Discrete Uniform distribution

Notation Used

Parameters

Probability Mass Function (PMF) and Support (Range)

Cumulative Distribution Function (CDF)

Expected Value (Mean)

Variance and Standard Deviation

Applications and Examples

Discrete Uniform Distribution: Equal-Chance Selection

The probabilistic experiment behind the discrete uniform distribution is one where a single outcome is selected from a finite set of possibilities, and every outcome is equally likely. There is no notion of success or failure, repetition, or accumulation — only a fair selection from a fixed range. Rolling a fair die or randomly choosing an integer from a known interval are typical examples. The defining feature of this experiment is the absence of bias: no outcome is favored over another.

The Probabilistic Experiment Behind Discrete Uniform distribution

The discrete uniform distribution represents situations where a random experiment produces outcomes from a finite set, and every outcome is equally likely. There is no preference, weighting, or bias toward any value in the set. The defining feature is not randomness alone, but symmetry: the mechanism that generates outcomes treats all possibilities identically.

This distribution is appropriate when the experiment consists of one selection from a known finite collection and there is no additional structure such as repeated trials, success–failure outcomes, or dependence between selections. Once the range of possible values is fixed, the distribution is fully determined. There are no hidden dynamics — probability is spread evenly across the entire support.

Due to this simplicity, the discrete uniform distribution often serves as a baseline model. More complex discrete distributions can frequently be understood as deviations from uniformity caused by repetition, conditioning, or structural constraints.

Learn More About Distinguishing Discrete Distributions →

Example:

Rolling a fair six-sided die once. The possible outcomes are

{1,2,3,4,5,6}

, and each value has the same probability. No outcome is favored over another by the mechanism of the experiment.

See More Examples of Discrete Uniform Distribution →

Notation Used

X \sim \text{Unif}(a, b)

X \sim \text{DU}(a, b)

— distribution of the random variable.

\text{DiscreteUniform}(a, b)

— used to denote the distribution itself (not the random variable).

\text{U}(a, b)

— also used, though it can refer to either discrete or continuous; context is important.

P(X = k) = \frac{1}{b - a + 1}, \quad \text{for } k = a, a+1, \ldots, b

— probability mass function (PMF), where:

a

— minimum value (lower bound)

b

— maximum value (upper bound)

k

— any integer in the range

[a, b]

b - a + 1

— total number of possible outcomes

Key properties:

E[X] = \frac{a + b}{2}

— expected value (mean)

\text{Var}(X) = \frac{(b - a + 1)^2 - 1}{12}

— variance

Special case:

\text{DiscreteUniform}(1, n)

— uniform distribution on

\{1, 2, \ldots, n\}

, often used for fair dice or simple random selection

See All Probability Symbols and Notations →

Parameters

a

: the smallest integer in the range

b

: the largest integer in the range

The uniform discrete distribution assigns equal probability to each integer between

a

and

b

, inclusive.

The values must be equally spaced and finite in number.

The parameters define the range — once

a

and

b

are set, every integer in that closed interval has probability

\frac{1}{b - a + 1}

.

This distribution is used when there's no reason to favor any outcome over another — every value is equally likely by design.

Probability Mass Function (PMF) and Support (Range)

The probability mass function (PMF) of a discrete uniform distribution is given by:

P(X = x) = \frac{1}{b - a + 1} = \frac{1}{n}, \quad x \in \{x_1, x_2, \dots, x_n\}

Where :

a

= lower bound (integer)

b

= upper bound (integer)

𝑛=b−a+1

is total number of possible values

Intuition Behind the Formula

Uniformity: The term "uniform" implies that each outcome is equally likely. That is, no single value of the random variable is preferred over another. This is the key feature of a uniform distribution.

Support (Range of the Random Variable):
* The random variable

X

can take on

n = b - a + 1

distinct values:

x_1, x_2, \ldots, x_n

.
* These values could be consecutive integers (like

1, 2, 3, \ldots, n

) or any set of

n

distinct values.
* The range or support is thus a finite, countable set.

Logic Behind the Formula:

The total probability must sum to 1:

\sum_{i=1}^n P(X = x_i) = 1

Since all probabilities are equal:

n \cdot \frac{1}{n} = (b - a + 1) \cdot \frac{1}{b - a + 1} = 1

This makes the individual probability of each outcome

\frac{1}{n} = \frac{1}{b - a + 1}

Discrete Uniform Distribution

Equal probability for finite outcomes

Minimum Value (a): 1

Maximum Value (b): 6

Explanation

A discrete uniform distribution assigns equal probability to each value in a finite range. The probability mass function is $P(X = k) = \frac{1}{b - a + 1}$ for $a \leq k \leq b$ . The expected value is $E[X] = \frac{a + b}{2}$ , and the variance is $\text{Var}(X) = \frac{n^2 - 1}{12}$ , where $n = b - a + 1$ . Common examples include rolling a fair die, selecting a random card from a deck, or generating a random number from a finite range.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a discrete uniform distribution is given by:

F_X(x) = P(X \leq x) = \begin{cases} 0, & x < a \\ \frac{\lfloor x \rfloor - a + 1}{b - a + 1}, & a \leq x \leq b \\ 1, & x > b \end{cases}

Where:

a

= lower bound (integer)

b

= upper bound (integer)

\lfloor x \rfloor

= floor function (largest integer less than or equal to

x

)

Intuition Behind the Formula

Definition: The CDF gives the probability that the random variable

X

takes on a value less than or equal to

x

.

Three Regions:

1. Before the range (

x < a

): No outcomes are possible below

a

, so

F_X(x) = 0

.

2. Within the range (

a \leq x \leq b

): The CDF counts how many integer values from

a

\lfloor x \rfloor

are included, divided by the total number of outcomes

(b - a + 1)

.
* Number of outcomes up to

\lfloor x \rfloor

\lfloor x \rfloor - a + 1

* Total outcomes:

b - a + 1

* Therefore:

F_X(x) = \frac{\lfloor x \rfloor - a + 1}{b - a + 1}

3. After the range (

x > b

): All outcomes have occurred, so

F_X(x) = 1

.

Step Function Behavior: Since

X

only takes integer values, the CDF remains constant between integers and jumps by

\frac{1}{b - a + 1}

at each integer point.

Discrete Uniform Distribution CDF

CDF rises uniformly in equal steps

Minimum Value (a): 1

Maximum Value (b): 6

CDF Explanation

The cumulative distribution function (CDF) for the discrete uniform distribution is

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

for

a \leq k \leq b

. The CDF increases in equal steps of

\frac{1}{n}

where

n = b - a + 1

is the number of possible values. Each step represents one additional outcome being included in the cumulative probability. The CDF reaches 1.0 at the maximum value

b

and remains at 1.0 for all larger values.

Expected Value (Mean)

As explained in the general case for calculating expected value, the expected value of a discrete random variable is computed as a weighted sum where each possible value is multiplied by its probability:

E[X] = \sum_{x} x \cdot P(X = x)

For the discrete uniform distribution, we apply this general formula to the specific probability mass function of this distribution.

Formula

E[X] = \frac{a + b}{2}

Where:

a

= minimum value (lower bound)

b

= maximum value (upper bound)

Derivation and Intuition

Starting from the general definition and substituting the PMF

P(X = x) = \frac{1}{b - a + 1}

for each

x \in \{a, a+1, \ldots, b\}

E[X] = \sum_{x=a}^{b} x \cdot \frac{1}{b - a + 1} = \frac{1}{b - a + 1} \sum_{x=a}^{b} x

The sum of integers from

a

b

is:

\sum_{x=a}^{b} x = \frac{(b - a + 1)(a + b)}{2}

Substituting back:

E[X] = \frac{1}{b - a + 1} \cdot \frac{(b - a + 1)(a + b)}{2} = \frac{a + b}{2}

The result

E[X] = \frac{a + b}{2}

captures the symmetry of the uniform distribution: the expected value is simply the midpoint between the minimum and maximum values. Since all outcomes are equally likely, the average naturally falls at the center of the range.

Example

Consider rolling a fair six-sided die, where

a = 1

and

b = 6

E[X] = \frac{1 + 6}{2} = 3.5

The expected value is 3.5, which is the arithmetic mean of all possible outcomes

\{1, 2, 3, 4, 5, 6\}

. Note that the expected value need not be a possible outcome of the random variable.

Variance and Standard Deviation

The variance of a discrete random variable measures how spread out the values are around the expected value. It is computed as:

\mathrm{Var}(X) = \mathbb{E}[(X - \mu)^2] = \sum_{x} (x - \mu)^2 P(X = x)

Or using the shortcut formula:

\mathrm{Var}(X) = \mathbb{E}[X^2] - \mu^2

For the discrete uniform distribution, we apply this formula to derive the variance.

Formula

\mathrm{Var}(X) = \frac{(b - a + 1)^2 - 1}{12}

Where:

a

= minimum value (lower bound)

b

= maximum value (upper bound)

Derivation and Intuition

Starting with the shortcut formula, we need to calculate

\mathbb{E}[X^2]

.

We know from the expected value section that

\mu = \frac{a + b}{2}

.

For

\mathbb{E}[X^2]

, using the PMF

P(X = x) = \frac{1}{b - a + 1}

\mathbb{E}[X^2] = \sum_{x=a}^{b} x^2 \cdot \frac{1}{b - a + 1} = \frac{1}{b - a + 1} \sum_{x=a}^{b} x^2

Using the formula for sum of squares:

\sum_{x=a}^{b} x^2 = \frac{(b - a + 1)(2b^2 + 2ab + 2a^2 - a - b)}{6}

After algebraic manipulation:

\mathrm{Var}(X) = \mathbb{E}[X^2] - \mu^2 = \frac{(b - a + 1)^2 - 1}{12}

The result shows that variance depends on the range width

(b - a + 1)

squared. A wider range of equally likely values produces higher variance. The formula is symmetric and increases quadratically with the spread between

a

and

b

Standard Deviation

\sigma = \sqrt{\mathrm{Var}(X)} = \sqrt{\frac{(b - a + 1)^2 - 1}{12}}

Example

For a fair six-sided die where

a = 1

and

b = 6

\mathrm{Var}(X) = \frac{(6 - 1 + 1)^2 - 1}{12} = \frac{36 - 1}{12} = \frac{35}{12} \approx 2.917

\sigma = \sqrt{\frac{35}{12}} \approx 1.708

The variance of approximately 2.917 indicates moderate spread around the mean of 3.5, with outcomes ranging from 1 to 6.

Applications and Examples

### Practical Example
Suppose you roll a fair six-sided die. The possible outcomes are

\{1, 2, 3, 4, 5, 6\}

, and the probability of each face is:

P(X = x) = \frac{1}{6} = \frac{1}{6 - 1 + 1}, \quad x = 1, 2, 3, 4, 5, 6

Each face has an equal and independent chance of appearing.