5.3: Discrete Uniform Distribution

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Discrete Uniform Distribution

A discrete uniform distribution is a probability distribution in which all outcomes are equally likely. This means each value in the distribution has the same probability of occurring. The distribution is called "discrete" because it applies to a finite number of specific outcomes (not a continuous range).

This distribution is common in settings where outcomes are balanced or randomized fairly. For example:

Rolling a fair six-sided die (each outcome 1 through 6 is equally likely).
Randomly selecting one of 10 raffle tickets.
Choosing a day of the week at random (Monday through Sunday).

The probability density function (pdf) of a discrete uniform distribution is

\[
P(X = x) = \frac{1}{n} \quad \text{for each } x \in \{x_1, x_2, \dots, x_n\}
\]

where:

n is the number of equally likely outcomes,
x is any one of those outcomes.

Similarly to the distributions we created when knowing each of the probabilities, theoretical probabilities calculated with formulas have an expected value.

The expected value is calculated by

\[
\mu = \frac{1}{n} \sum_{i=1}^{n} x_i
\]

Example 1: Random Integer from 1 to 5

Suppose a research assistant randomly selects an integer from 1 to 5 to assign each of five survey versions to participants.

a) What is the probability of selecting version 3?
b) What is the expected value of the selected version?

Solution:

There are 5 values: 1, 2, 3, 4, 5.
This is a discrete uniform distribution with a=1a = 1a=1, b=5b = 5b=5.

a) What is the probability of selecting version 3?

\[
P(X = 3) = \frac{1}{5} = 0.2
\]

b) What is the expected value of the selected version? For consecutive values you can use this shortcut formula.

\[
\mu = \frac{a + b}{2} = \frac{1 + 5}{2} = 3
\]

Example 3: Discrete Uniform in Economics

Suppose an economist models a simplified version of job search behavior where a worker receives one job offer per week and each offer has a uniformly distributed wage between $15 and $25 per hour (in $1 increments).

a) What is the probability that an offer is exactly $20/hour?
b) What is the expected wage offer?

Solution:

Possible wages: $15, $16, ..., $25 → 11 values

This is a discrete uniform distribution with:
\[
a = 15, \quad b = 25, \quad n = b - a + 1 = 11
\]

a) \[
P(X = 20) = \frac{1}{11} \approx 0.0909
\]

b) \[
\mu = \frac{1}{n} \sum_{i=1}^{n} x_i
= \frac{1}{11}(15 + 16 + 17 + \dots + 25)
= \frac{1}{11}(220) = 20
\]

c) Wages less than $18 include: $15, $16, $17 — 3 outcomes out of 11.

\[
P(X < 18) = P(X = 15) + P(X = 16) + P(X = 17)
= \frac{1}{11} + \frac{1}{11} + \frac{1}{11} = \frac{3}{11} \approx 0.2727
\]

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