Moment Generating Functions

The moment generating function packs an entire distribution's moment structure into one function of a dummy variable $t$, and then makes two hard operations, computing high moments, and finding the law of a sum of independent variables, into differentiation and multiplication. It is the algebraic engine behind clean proofs of the The Central Limit Theorem and the reason certain distribution families are closed under addition. Master it and a class of otherwise painful problems collapses to bookkeeping.

Definition and the "generating" property

The moment generating function (MGF) of a random variable $X$ is $M_X(t) = \mathbb{E}\big[e^{tX}\big],$ defined for $t$ in some neighbourhood of $0$ where the expectation is finite. The name comes from expanding the exponential and using linearity of Expectation, Variance & Moments: $M_X(t) = \mathbb{E}\!\left[\sum_{k=0}^\infty \frac{(tX)^k}{k!}\right] = \sum_{k=0}^\infty \frac{t^k}{k!}\,\mathbb{E}[X^k].$ So $M_X$ is the exponential generating function of the moments: the $k$-th moment is the $k$-th derivative at zero, $\mathbb{E}[X^k] = M_X^{(k)}(0).$ In particular $M_X(0) = 1$, $M_X'(0) = \mathbb{E}[X]$, and $M_X''(0) = \mathbb{E}[X^2]$, so $\operatorname{Var}(X) = M''_X(0) - M'_X(0)^2$. One function, differentiated repeatedly at a point, yields every moment.

Uniqueness: the MGF determines the law

The reason the MGF is a characterisation and not just a summary: if $M_X(t) = M_Y(t)$ for all $t$ in an open interval around $0$, then $X$ and $Y$ have the same distribution. This is what licenses the standard proof technique, compute an MGF, recognise it, and conclude the distribution. The caveat is existence: the MGF requires $e^{tX}$ to have finite expectation near $0$, which fails for heavy-tailed laws (lognormal, Student-$t$, Cauchy) whose moments blow up. When the MGF does not exist, one uses the characteristic function $\varphi_X(t) = \mathbb{E}[e^{itX}]$, which always exists ($|e^{itX}| = 1$), retains the uniqueness property, and is the tool serious proofs actually use. The MGF is the friendlier version that works when tails are light.

The multiplication rule for independent sums

Here is the property that makes MGFs indispensable. If $X$ and $Y$ are independent, then $M_{X+Y}(t) = \mathbb{E}\big[e^{t(X+Y)}\big] = \mathbb{E}\big[e^{tX}\big]\,\mathbb{E}\big[e^{tY}\big] = M_X(t)\,M_Y(t),$ the middle step using that independence factorises the expectation of a product. Convolution becomes multiplication. Finding the density of a sum directly requires an integral convolution; multiplying MGFs and recognising the result is far easier. This immediately explains why several distribution families are closed under addition, the Common Distributions relationships fall out mechanically:

• Two independent normals: $M(t) = e^{\mu_1 t + \sigma_1^2 t^2/2}\,e^{\mu_2 t + \sigma_2^2 t^2/2} = e^{(\mu_1+\mu_2)t + (\sigma_1^2+\sigma_2^2)t^2/2}$, normal, means and variances add.
• Two independent Poissons: $e^{\lambda_1(e^t-1)}\,e^{\lambda_2(e^t-1)} = e^{(\lambda_1+\lambda_2)(e^t-1)}$, Poisson with rate $\lambda_1 + \lambda_2$.
• $n$ independent Bernoulli($p$): $(pe^t + q)^n$, the Binomial MGF, proving the sum is Binomial.

Cumulants: the additive moments

Take logs of the MGF to get the cumulant generating function: $K_X(t) = \log M_X(t) = \sum_{n=1}^\infty \kappa_n \frac{t^n}{n!}.$ The coefficients $\kappa_n$ are the cumulants, and they are the "right" descriptors of shape because the multiplication rule becomes addition under the log: for independent $X, Y$, $K_{X+Y}(t) = K_X(t) + K_Y(t) \;\Longrightarrow\; \kappa_n(X+Y) = \kappa_n(X) + \kappa_n(Y).$ Cumulants are additive over independent sums. The low ones are interpretable: $\kappa_1 = \mu, \quad \kappa_2 = \sigma^2, \quad \kappa_3 = \mathbb{E}[(X-\mu)^3], \quad \kappa_4 = \mathbb{E}[(X-\mu)^4] - 3\sigma^4,$ so $\kappa_3/\sigma^3$ is skewness and $\kappa_4/\sigma^4$ is excess kurtosis. The normal's signature is that all cumulants beyond the second vanish, $K(t) = \mu t + \tfrac12\sigma^2 t^2$ is quadratic, nothing higher. This is the sharpest possible statement of "the normal has no skew and no excess kurtosis", and it is why cumulants are the natural language for measuring departures from normality in return data.

Worked example, CLT scaling made obvious

Cumulant additivity gives the The Central Limit Theorem's normalisation in one line. Let $X_1,\dots,X_n$ be i.i.d. with mean $0$, variance $\sigma^2$, and cumulants $\kappa_n$. For the standardised sum $S_n = \frac{1}{\sqrt n}\sum X_i$, cumulants scale as $\kappa_m(S_n) = n^{1-m/2}\kappa_m$. So $\kappa_2(S_n) = \sigma^2$ stays fixed, while for $m \ge 3$, $\kappa_m(S_n) = n^{1-m/2}\kappa_m \to 0$. Every cumulant above the second is killed by the $\sqrt n$ scaling, leaving only the mean and variance, precisely a normal. The Edgeworth expansion formalises the leading correction: for finite $n$ the density picks up a skewness term of order $\kappa_3/\sqrt n$, which is exactly why small samples of skewed data are noticeably non-normal and why the The Central Limit Theorem convergence is slow in the tails.

Failure modes and subtleties

• Non-existence under heavy tails. If $\mathbb{E}[e^{tX}] = \infty$ for all $t > 0$ (lognormal, Student-$t$), the MGF simply does not exist; switch to the characteristic function. This is not a technicality in finance, where fat tails are the norm.
• Moments existing does not guarantee the MGF exists, and even when all moments are finite the moment sequence need not determine the law uniquely (the lognormal is the classic moment-indeterminate example, different distributions share all its moments). Uniqueness needs the MGF/CF, not the moments alone.
• Independence is required for multiplication. $M_{X+Y} = M_X M_Y$ fails for correlated variables; the cross terms in $\mathbb{E}[e^{t(X+Y)}]$ do not factor.
• Cumulants above two are not additive under scaling, they shrink. This is a feature (it drives the CLT) but a trap if you forget the $n^{1-m/2}$ factor when combining.

In interviews

The MGF is prized as a fast route to moments: "use the MGF to find the mean and variance of a Poisson", differentiate $e^{\lambda(e^t-1)}$ twice at $0$. Be ready to prove that a sum of independent Poissons (or normals) stays in the family via the multiplication rule, a clean two-line answer that signals fluency. A strong follow-up is "why might the MGF not exist, and what do you use instead?", name heavy tails and the characteristic function. The deepest point you can make is that a normal is exactly the distribution whose cumulant generating function is quadratic, which reframes both the The Central Limit Theorem and every "test for normality" as a statement about cumulants beyond the second.