Modeling With Discrete Uniform Distributions

Apply the discrete uniform distribution to model real-world scenarios with finitely many equally likely integer outcomes. Compute probabilities of events and intervals, and find expected values in context.

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Modeling With the Discrete Uniform Distribution

When an experiment produces a finite set of integer outcomes that are all equally likely, we model it with a discrete uniform distribution. Writing $X \sim \text{DU}(a, b)$ means $X$ takes each integer value in $\{a, a+1, \ldots, b\}$ with equal probability, and the PMF is

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

To build the model from a real situation:

Identify the set of outcomes and verify they are equally likely.
Find the smallest outcome $a$ and the largest outcome $b$ .
The total number of outcomes is $n = b - a + 1$ , so every outcome has probability $\dfrac{1}{n}$ .

Common situations that fit this model include rolling a fair $n$ -sided die, drawing a numbered ticket where each ticket is equally likely, and randomly generating an integer code in a fixed range.

For instance, a fair $20$ -sided die with faces $1$ through $20$ produces $X \sim \text{DU}(1, 20)$ , and

P(X = 7) = \frac{1}{20}.

To find the probability of an event $A \subseteq \{a, \ldots, b\}$ , count the outcomes in $A$ and divide by $n$ :

P(X \in A) = \frac{|A|}{n}.