The uniform distribution

All values equally likely

The uniform distribution is the model for "no value is more likely than any other" within a range. It comes in two forms: the discrete uniform for the whole numbers from \(a\) to \(b\), and the continuous uniform for any value between \(a\) and \(b\).

The discrete uniform

Each of the \(b - a + 1\) whole numbers from \(a\) to \(b\) has the same probability:

A die is the discrete uniform from 1 to 6. A bookseller who sells between 90 and 100 books a day, every total equally likely, has 11 possible totals, each with probability \(\tfrac{1}{11} = 0.0909\). The mean is 95 books, the variance \(\tfrac{11^2 - 1}{12} = 10\) and the standard deviation 3.1623 books.

For a count the boundary matters. At least 96 books means 96, 97, 98, 99 or 100: \(P(X \ge 96) = \tfrac{5}{11} = 0.4545\). More than 96 leaves 96 out: \(P(X > 96) = \tfrac{4}{11} = 0.3636\). Fewer than 93 means 90, 91 or 92: \(P(X < 93) = \tfrac{3}{11} = 0.2727\).

A common mistake: a continuous model for a count

It is tempting to model the bookseller with the continuous uniform from 90 to 100: uniform.cdf(95, loc=90, scale=10) gives 0.5. But sales are whole numbers, and 6 of the 11 equally likely totals are 95 or fewer, so \(P(X \le 95) = \tfrac{6}{11} = 0.5455\). The continuous model loses the probability that sits on each whole number, and it cannot tell "at least 96" from "more than 95". Use a true discrete uniform, scipy's randint(90, 101), whose upper end is exclusive.

The continuous uniform

For a measurement that is equally likely to fall anywhere between \(a\) and \(b\), the density is flat at \(\tfrac{1}{b - a}\), and a probability is a length divided by the width:

A software team finds that fixing a bug takes anywhere from 1 to 5 hours, no time more likely than another: \(X \sim U(1, 5)\), with a density of \(\tfrac{1}{4} = 0.25\) per hour. The chance that a fix takes more than 4 hours is \(\tfrac{5 - 4}{4} = 0.25\), and between 2 and 3.5 hours \(\tfrac{1.5}{4} = 0.375\). The mean is 3 hours, the standard deviation \(\tfrac{4}{\sqrt{12}} = 1.1547\) hours, and 90 % of fixes take at most \(1 + 0.9 \times 4 = 4.6\) hours.

For a measurement the boundary makes no difference: \(P(X < 3) = P(X \le 3) = 0.5\), because a single point has no width and so no probability.

A density can be greater than 1

If fixes took between 1 and 1.5 hours, the density would be \(\tfrac{1}{0.5} = 2\) per hour. That is not a probability of 2: it is probability per hour, over a range half an hour wide, and the area under it is still \(2 \times 0.5 = 1\). That is why the axis of a density is labelled "density", never "probability".

When "all equally likely" is not a fair model

• Real sales cluster. Most shops have typical days and rare extremes. If the bookseller's records show many days near 95 and few at 90 or 100, the uniform overstates the extremes.
• Ends that are not really ends. The uniform puts no probability outside \([a, b]\). If a bug can now and then take 8 hours, \(U(1, 5)\) says that never happens.
• Schedules. The wait for a bus that comes every 10 minutes is uniform only for someone who arrives at a random moment; a commuter who knows the timetable waits less.
• Where it fits: fair games of chance, random numbers (every simulation starts from \(U(0, 1)\)), and an honest "we know the range and nothing more".

In code

In scipy, randint(a, b + 1) is the discrete uniform from a to b, and uniform(loc=a, scale=b − a) the continuous one. Excel has no uniform functions, so the formulas are the arithmetic itself.