Discrete Uniform Random Variables

A raffle draws an integer from 5 to 20, each equally likely. Its variance is 21.25. Reach for the continuous uniform formula and you get 18.75, off by 12%. The discrete uniform has its own variance formula, and the difference is tested.

A discrete uniform random variable takes each integer in $\{a, a+1, \ldots, b\}$ with equal probability. Let $k = b - a + 1$ be the number of values.

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

The mean is the midpoint, and the variance depends only on the count of values:

$$E[X] = \frac{a+b}{2}, \quad \text{Var}(X) = \frac{k^2 - 1}{12}$$

A clean way to find the variance is to shift to the standard set $\{1, 2, \ldots, k\}$, which has the same spread; shifting changes the mean but never the variance.

HIGH-FREQUENCY: The discrete uniform variance is $(k^2-1)/12$ with $k$ the COUNT of values, not the continuous $(b-a)^2/12$.

Common mistakes

• Using $(b-a)^2/12$ for the discrete uniform. That is the continuous formula. The discrete variance is $(k^2-1)/12$ with $k$ the count of values. For $\{5,\ldots,20\}$: 21.25, not 18.75.
• Miscounting $k$. The set $\{a, \ldots, b\}$ has $b - a + 1$ values, not $b - a$. For $\{5,\ldots,20\}$, $k = 16$, not 15.
• Scaling variance by $c$ instead of $c^2$. $\text{Var}(cX) = c^2\text{Var}(X)$. For $X = 100N$, multiply the variance by 10,000, not 100.

Bottom line

• Discrete uniform on $\{a, a+1, \ldots, b\}$: $k = b - a + 1$ equally likely values, each with probability $1/k$.
• Mean: $E[X] = (a+b)/2$, the midpoint, regardless of the count $k$.
• Variance: $\text{Var}(X) = (k^2 - 1)/12$, where $k$ is the COUNT of values.
• Watch the trap: the continuous uniform uses $(b-a)^2/12$. These differ.

Exam shortcut

The instant you see "equally likely integers," write down $k = b - a + 1$, then $E = (a+b)/2$ and $\text{Var} = (k^2-1)/12$. The trap is always the continuous formula sneaking in. "Count the values, then $(k^2-1)/12$." Discrete uses the count; continuous uses the width. "Midpoint mean" holds for both.

The full lesson (about 1,460 words, 10 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• 2a
• 2b
• 2c
• 2d
• 2e
• 2f

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