Why MGF Differentiation Is Valid

Why Differentiating an MGF Needs Justification

The moment-generating function of a random variable \(X\) is \(M_X(t)=E[e^{tX}]\) wherever this expectation is finite. It is tempting to differentiate the exponential inside the expectation and write \(M_X'(t)=E[Xe^{tX}]\). Pointwise, the derivative of \(e^{tX}\) is indeed \(Xe^{tX}\). But pointwise differentiation alone does not justify interchanging differentiation and expectation: the difference quotients must be controlled by an integrable random variable.

Finiteness on an open neighborhood provides exactly the extra control needed. The exponential factor can absorb powers of \(|X|\), and a slightly larger exponential moment bounds the difference quotients. The result is a derivative formula valid not just for the first derivative, but for every finite order.

The Exponential Bound Behind the Formula

The key inequality is that an exponential with a little extra rate dominates any fixed power. For \(c>0\) and a nonnegative integer \(r\), there is a finite constant \(C_{r,c}\) such that

For example, when \(x\ge 0\), the function \(x^r e^{-cx}\) is bounded: it is continuous and tends to zero as \(x\) tends to infinity. The same reasoning applies to \(|x|\) when \(x<0\). Thus an exponential moment with a margin beyond \(t\) controls both powers of \(X\) and the nearby difference quotients.

Here is the first-order bound. For \(h\ne 0\), the mean value theorem applied to the exponential gives

Multiplying by \(e^{tx}\), and choosing \(|h|\) smaller than the available neighborhood margin, produces an integrable bound. This is what permits the dominated convergence theorem to be used.

In particular, if the MGF is finite on an open interval around zero, then \(E[X^r]\) exists absolutely for every nonnegative integer \(r\), and \(M_X^{(r)}(0)=E[X^r]\).

Proof of the Derivative Formula

We prove the result locally around an arbitrary \(t_0\) in the interval of finiteness. Choose \(\delta>0\) small enough that \(M_X(t_0+2\delta)\) and \(M_X(t_0-2\delta)\) are finite. Define

for \(s\) sufficiently close to \(t_0\). First, these expectations exist absolutely. For \(|s-t_0|\le\delta\), the exponential bound on powers gives a constant \(C\), depending on \(r\) and \(\delta\), such that

To see the two sides of this bound, when \(x\ge0\) use the term with exponent \(t_0+2\delta\); when \(x<0\) use the term with exponent \(t_0-2\delta\). The right-hand side has finite expectation by the choice of \(\delta\).

Now fix \(s\) with \(|s-t_0|<\delta\). For sufficiently small \(|h|\), the difference quotient for \(g_r\) satisfies

where the second inequality follows by absorbing \(|X|^{r+1}\) into the remaining exponential margin. The final random variable is integrable. For each fixed value of \(X\), the difference quotient converges to \(X^{r+1}e^{sX}\) as \(h\to0\). The dominated convergence theorem therefore yields

Since \(g_0(s)=M_X(s)\), repeated application of this identity gives \(M_X^{(r)}(s)=g_r(s)\) for every finite \(r\). In particular, the conclusion holds at \(t_0\). When \(t_0=0\), \(e^{0X}=1\), so \(M_X^{(r)}(0)=E[X^r]\). The same exponential bounds also show \(E[|X|^r]<\infty\). \(\square\)

Worked Examples

Why the Neighborhood Condition Matters

The condition is more than a technical detail. Although \(M_X(0)=E[1]=1\) for every random variable, that fact alone does not provide an integrable bound for difference quotients near zero. The proof uses finite MGF values at points on both sides of the point where differentiation is being performed. Those values provide the margin that controls the tails of \(X\).

• Common mistake: “The derivative of \(e^{tX}\) is \(Xe^{tX}\), so \(M_X'(t)=E[Xe^{tX}]\) automatically.” This applies pointwise, but does not establish that the limit can pass through the expectation.
• Correction: Check that the MGF is finite on an open neighborhood of \(t\), then use an integrable exponential bound on the difference quotients and the dominated convergence theorem.
• At zero: A finite MGF on a neighborhood of zero guarantees all ordinary moments exist and can be recovered by derivatives there. Finiteness only at zero does not establish that conclusion.

The formula is useful because it connects an analytic object, the MGF, to probabilistic quantities, the moments. The neighborhood condition explains why this connection is valid rather than merely plausible. It also signals the limit of the method: before differentiating, check the domain and verify that the point lies in its interior.