Maths › Further Statistics 2 › The continuous uniform distribution
The continuous uniform distribution
A flat density on an interval, which sounds too simple to be worth a name. It is the model for rounding error and waiting time, and its mean and variance come out of the general formulae in three lines.
Builds on Mean, variance and skewness and Density and distribution functions.
IN THIS TOPIC
- Write down the density and distribution function of U(a, b).
- Derive its mean and variance from the general integrals.
- Recognise situations the model fits, and read probabilities off as lengths.
WHAT YOU PROBABLY THINK
The variance of the uniform distribution on a to b is (b − a)²/4, since the range is b − a.
A rectangle of area one
If X is equally likely to fall anywhere in [a, b] then the density must be flat, and its area must be 1, so its height is 1/(b − a). Integrating gives a straight ramp for the distribution function:
Probabilities are therefore proportions of length: P(c < X < d) is (d − c)/(b − a) for any interval inside the range. Rounding to the nearest unit gives an error modelled by U(−0.5, 0.5), and a wait for a service that arrives every T minutes is modelled by U(0, T).
WORKED EXAMPLE
Reading off a uniform model
X follows U(2, 8). Find P(3 < X < 5) and P(X > 7).
The height is 1/6. P(3 < X < 5) = 2 × 1/6 = 1/3.
P(X > 7) = 1 × 1/6 = 1/6. No integration was needed: with a flat density, lengths do all the work.
Mean and variance, derived
The mean is the midpoint, which is obvious from the symmetry and confirmed by the integral. The variance takes one more line:
The 12 is the part worth remembering, and the opening claim gets it wrong: the range is b − a, but the spread about the mean is far smaller than the range itself, and dividing by 4 would make the standard deviation half the range rather than about 29% of it.
YOUR TURN
Deriving the variance
For X following U(a, b), show that Var(X) = (b − a)²/12, and evaluate the mean and standard deviation for U(2, 8).
Show the working
E(X²) = ∫ x²/(b − a) dx from a to b = (b³ − a³)/[3(b − a)] = (a² + ab + b²)/3, using the difference of cubes.
Subtracting [(a + b)/2]² = (a² + 2ab + b²)/4 gives (4a² + 4ab + 4b² − 3a² − 6ab − 3b²)/12 = (a² − 2ab + b²)/12.
That is (b − a)²/12, as required.
For U(2, 8): mean 5, variance 36/12 = 3, and standard deviation √3 ≈ 1.73.
THE EXAM BIT
- State the height of the density as 1/(b − a) before using it; marks are given for it.
- Write F(x) piecewise, with 0 below a and 1 above b as well as the ramp between them.
- For probabilities, use lengths rather than integrating: it is faster and harder to slip.
- The variance divides by 12; check it against the rough rule that the standard deviation is about 29% of the range.
CHECK YOURSELF
X follows U(0, 10). Find P(X > 6.5) and the standard deviation.
Show a hint
A length over the range, then the variance formula.
Show the answer
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U(a, b) has density 1/(b − a) and distribution function (x − a)/(b − a), so probability is proportion of length.
Its mean is the midpoint (a + b)/2 and its variance is (b − a)²/12, giving a standard deviation of about 29% of the range.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Write down the density and distribution function of U(a, b).
- Derive its mean and variance from the general integrals.
- Recognise situations the model fits, and read probabilities off as lengths.
Open the full revision checklist to see every objective in the course in one place.
No animated video for this topic yet; these notes stand alone.