Discrete Uniform Random Variables | Free Exam P Lesson

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 ${a, a + 1, \dots, b}$ with equal probability. Let $k = b - a + 1$ be the number of values.

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

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

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

A clean way to find the variance is to shift to the standard set ${1, 2, \dots, k}$ , 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 $(b - a)^{2} /12$ .

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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 ${5, \dots, 20}$ : 21.25, not 18.75.
Miscounting $k$ . The set ${a, \dots, b}$ has $b - a + 1$ values, not $b - a$ . For ${5, \dots, 20}$ , $k = 16$ , not 15.
Scaling variance by $c$ instead of $c^{2}$ . $Var (c X) = c^{2} Var (X)$ . For $X = 100 N$ , multiply the variance by 10,000, not 100.

Bottom line

Discrete uniform on ${a, a + 1, \dots, b}$ : $k = b - a + 1$ equally likely values, each with probability $1/ k$ .
Mean: $E [X] = (a + b) /2$ , the midpoint, regardless of the count $k$ .
Variance: $Var (X) = (k^{2} - 1) /12$ , 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 $Var = (k^{2} - 1) /12$ . 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.

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Learning objectives

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