Normal Moments and Symmetry from the MGF

Why the Normal MGF Is Especially Useful

For a normal random variable, the moment-generating function has a simple exponential form. Centering the variable makes that form an even function of its argument, revealing a precise pattern in its moments: all odd central moments are zero, while the even central moments follow a recurrence. These facts are useful both for calculating moments efficiently and for recognizing which features of a distribution are determined by its mean and variance.

We derive the MGF directly from the normal density, then use differentiation to establish the moment pattern. Throughout, let \(X\sim N(\mu,\sigma^2)\), where \(\mu\in\mathbb{R}\) and \(\sigma>0\), and define the centered variable \(Z=X-\mu\).

We will use the MGF differentiation result established in the earlier material on recovering moments by differentiating an MGF: if an MGF is finite on an open interval containing zero, then its \(r\)th derivative at zero exists and equals \(E[X^r]\), provided the corresponding moment is considered. The normal MGF will be finite for every real \(t\), so this condition holds for every derivative order used below.

Deriving the Normal MGF

The density of \(X\sim N(\mu,\sigma^2)\) is \(f(x)=\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)\). Starting with the expectation integral, complete the square in the exponent:

For the second equality, the relevant algebra is \(-\frac{(x-\mu)^2}{2\sigma^2}+tx =-\frac{(x-\mu-\sigma^2t)^2}{2\sigma^2}+\mu t+\frac{\sigma^2t^2}{2}\). The remaining integral is one: its integrand is a normal density with mean \(\mu+\sigma^2t\) and variance \(\sigma^2\). Thus the integral is finite for every real \(t\).

Why Odd Central Moments Vanish

The MGF of the centered variable is \(M_Z(t)=\exp(\sigma^2t^2/2)\). Replacing \(t\) with \(-t\) leaves it unchanged, so this MGF is even. Since the normal MGF is finite for all real arguments, it can be differentiated to every order at zero. The derivatives of odd order of an even function at zero are zero; applying the MGF differentiation result therefore gives the odd central moments.

To justify the derivative claim directly, if a function \(g\) is even and has an \(r\)th derivative at zero, then differentiating \(g(-t)=g(t)\) \(r\) times and setting \(t=0\) gives \((-1)^r g^{(r)}(0)=g^{(r)}(0)\). For odd \(r\), this implies \(g^{(r)}(0)=0\). Taking \(g=M_Z\), the derivative-MGF identity yields \(E[Z^r]=M_Z^{(r)}(0)=0\). Since \(Z=X-\mu\), these are exactly the odd central moments of \(X\). \(\square\)

This conclusion concerns moments about the mean, not necessarily raw moments about zero. If \(\mu\ne 0\), an odd raw moment such as \(E[X^3]\) need not vanish. The mean shift changes raw moments even though the centered MGF remains even.

A Recurrence for the Even Central Moments

The even central moments can be obtained without repeatedly expanding the exponential. Differentiate the centered MGF: \(M_Z'(t)=\sigma^2tM_Z(t)\). Write \(m_r=E[Z^r]=M_Z^{(r)}(0)\), with \(m_0=1\). For any integer \(r\geq 1\), differentiate this identity \(r\) times. By the product rule, the \(r\)th derivative of \(tM_Z(t)\) is \(tM_Z^{(r)}(t)+rM_Z^{(r-1)}(t)\). Evaluating at zero removes the term containing \(t\), giving the recurrence below.

For completeness, the recurrence gives \(m_2=\sigma^2m_0=\sigma^2\), \(m_4=3\sigma^2m_2=3\sigma^4\), and in general \(m_{2k}=(2k-1)\sigma^2m_{2k-2}\). Iterating from \(m_0=1\) yields \(m_{2k}=(2k-1)(2k-3)\cdots 3\cdot1\,\sigma^{2k}\). The product of the positive odd integers through \(2k-1\) equals \((2k)!/(2^k k!)\): the even factors in \((2k)!\) multiply to \(2^k k!\), leaving precisely the odd factors. This proves the closed form. The odd moments are zero by the evenness argument above. \(\square\)

Raw Moments Depend on the Mean

The recurrence is a direct method for central moments. Raw moments can be found by differentiating \(M_X(t)\) itself, or by writing \(X=\mu+Z\) and expanding. In the latter approach, the centered moments already established determine which terms remain: odd powers of \(Z\) have expectation zero, and even powers use the formula above.

Common Pitfall and Key Takeaway

A common mistake is to infer that every odd raw moment of a normal variable is zero because the centered MGF is even. The evenness applies to \(M_Z(t)\), where \(Z=X-\mu\), not generally to \(M_X(t)\). For example, the preceding calculation gives \(E[X^3]=-13\) for a normal variable with mean \(-1\), even though its third central moment is zero. Always identify whether the moment is about zero or about the mean before applying the symmetry result.

The central moment formulas provide a compact description of the normal distribution's moment pattern: the mean sets its location, the variance determines every central moment, and all odd central moments vanish. These calculations are also a useful check when deriving raw moments or verifying expressions obtained by direct differentiation.