Raw and Central Moments of a Uniform Variable

Why Moments Are Useful

A distribution can be described in several ways. Probabilities and quantiles answer questions about events and cutoffs; moments summarize aspects of the distribution's location, spread, and shape. For a uniform distribution on a finite interval, the density is constant, so these summaries can be derived directly with elementary integrals. The calculations also give a useful example of the distinction between a moment about zero and a moment about the mean.

Let \(X\sim\operatorname{Unif}(a,b)\), where \(a<b\) are finite real numbers. Its density is \(f(x)=1/(b-a)\) for \(a<x<b\), and zero elsewhere. Values of a density at the endpoints do not affect the integrals below. Since \(X\) is bounded, every nonnegative integer moment considered here exists and is finite.

Raw Moments by Direct Integration

For any nonnegative integer \(k\), the expectation integral uses the density on the interval \((a,b)\). Integrating the power \(x^k\) gives a formula for all raw moments at once.

This formula includes \(k=0\), for which it gives \(E[1]=1\), as it must. For \(k=1\), factoring \(b^2-a^2=(b-a)(a+b)\) gives the mean. For \(k=2\), it gives the raw second moment.

Mean and Variance

The mean is the first raw moment, \(\mu=(a+b)/2\), the midpoint of the interval. To obtain the variance from raw moments, first expand its definition:

The last equality uses \(E[X]=\mu\). Substituting the raw second moment and the mean yields

There is also a direct centered-integral derivation. It is useful for obtaining the higher central moments, so we develop it more generally. Set \(m=(a+b)/2\) and \(d=b-a>0\). In the integral for a central moment, use \(y=x-m\). Then \(x=a\) corresponds to \(y=-d/2\), \(x=b\) corresponds to \(y=d/2\), and the density remains \(1/d\). Thus

Higher Central Moments

The centered integration formula immediately explains a symmetry property. For odd \(k\), the function \(y^k\) is odd, and the integration limits are symmetric about zero. Its integral is therefore zero. For even \(k\), integrate the power over the positive and negative halves, whose contributions are equal.

Proof. The centered integral is \(d^{-1}\int_{-d/2}^{d/2}y^k\,dy\). If \(k\) is odd, an antiderivative is \(y^{k+1}/(k+1)\), and \(k+1\) is even. The values at \(d/2\) and \(-d/2\) are equal, so their difference is zero.

If \(k\) is even, the integrand is even, and hence \[ \frac{1}{d}\int_{-d/2}^{d/2}y^k\,dy =\frac{2}{d}\int_0^{d/2}y^k\,dy =\frac{2}{d}\cdot\frac{(d/2)^{k+1}}{k+1} =\frac{d^k}{2^k(k+1)}. \] Both cases follow from the density integral, so no additional symmetry assumption is needed beyond the uniform density on the interval. \(\square\)

For \(k=1\), the result confirms that the first central moment is zero. For \(k=2\), it recovers the variance \(d^2/12\). The next even central moments are

The third central moment is zero. When standardized moments are desired, the standardized third central moment is \(E[(X-\mu)^3]/\operatorname{Var}(X)^{3/2}=0\). The standardized fourth central moment is \[ \frac{E[(X-\mu)^4]}{\operatorname{Var}(X)^2} =\frac{d^4/80}{(d^2/12)^2} =\frac{d^4/80}{d^4/144} =\frac{9}{5}. \] This ratio is sometimes called the kurtosis; the excess kurtosis, defined as kurtosis minus \(3\), is \(-6/5\). These ratios do not depend on the interval endpoints because the powers of its width cancel.

Common Mistakes and Checks

Moments are compact summaries, but their formulas are easy to misapply. A reliable calculation starts by identifying both the power and the center about which it is taken.

• Using the raw second moment as the variance. The false step is to write \(\operatorname{Var}(X)=E[X^2]\). The definition instead gives \(\operatorname{Var}(X)=E[X^2]-(E[X])^2\). For \(X\sim\operatorname{Unif}(2,8)\), these values are \(28\) and \(3\), respectively.
• Centering at zero when a central moment is requested. \(E[X^4]\) and \(E[(X-\mu)^4]\) are different quantities in general. A central moment requires subtracting the mean inside the power before taking the expectation.
• Assuming every odd raw moment is zero. It is odd central moments that vanish here, because the interval is symmetric about its mean. If the interval is shifted away from zero, an odd raw moment need not vanish.
• Forgetting the interval width in the density. The density is \(1/(b-a)\), not \(1\) unless the interval has width \(1\). Omitting this factor produces an integral that is incorrectly scaled.

A quick check is to confirm that the mean lies halfway between the endpoints, the variance is positive and scales with the square of the interval width, and every odd central moment is zero. The second-moment formula also gives a useful consistency check: substituting the mean and raw second moment must reproduce \((b-a)^2/12\).