Moment Generating Functions

Two distributions have the same mean and variance but differ in the third moment. The MGF captures all moments at once, and if two random variables share the same MGF, they have the same distribution. Period.

The moment generating function of a random variable $X$ is:

$$M_X(t)=E[e^{tX}]$$

For a discrete random variable: $M_X(t)={\sum }_x{e}^{tx}{p}_X(x)$. For a continuous random variable: $M_X(t)={\int }_{-\infty }^{\infty }{e}^{tx}{f}_X(x)dx$.

The MGF must converge for all $t$ in some open interval containing 0. If the integral diverges for all $t\ne 0$, the MGF does not exist. Heavy-tailed distributions like the Cauchy and lognormal have no MGF.

HIGH-FREQUENCY: The MGF identifies the distribution uniquely. If you can compute $M_X(t)$ and recognize its form, you know the distribution, no further work needed.

Differentiate the MGF and evaluate at $t=0$:

$$E[X^k]=M_X^{(k)}(0)=\frac{d^k}{d{t}^k}{M}_X(t){|}_{t=0}$$

Common mistakes

• Forgetting to evaluate at $t=0$ after differentiating. The first moment is $M^{\prime }(0)$, not $M^{\prime }(t)$. Leaving $t$ in the expression gives a function, not a number. Trap: reporting an expression in $t$ as the expected value.
• Adding MGFs instead of multiplying them. The MGF of a sum of independents is the product of the MGFs, not the sum. $M_{X+Y}(t)=M_X(t)\cdot M_Y(t)$. Adding gives a meaningless quantity.
• Confusing scale and rate parametrizations. The exponential MGF is $(1-\theta t{)}^{-1}$ with scale $\theta$ (mean = $\theta$), or $\lambda /(\lambda -t)$ with rate $\lambda$. Mixing them produces the wrong moments. Trap: using mean 2 for Gamma($3,2$) when scale = 2 means mean = 6.

Bottom line

• MGF definition: $M_X(t)=E[e^{tX}]$, and it must exist in a neighborhood of $t=0$
• k-th moment: $E[X^k]=M_X^{(k)}(0)$, differentiate k times then evaluate at zero
• Variance shortcut: $\text{Var}(X)=M^{\prime \prime }(0)-[M^{\prime }(0){]}^2$, faster than the raw definition
• Uniqueness: if $M_X(t)=M_Y(t)$ for all $t$ near 0, then $X$ and $Y$ share the same distribution

Exam shortcut

When you see an unfamiliar MGF, compare it to the standard forms: $e^{\mu t+{\sigma }^2{t}^2/2}$ (normal), $(1-\theta t{)}^{-\alpha }$ (gamma), $e^{\lambda (e^t-1)}$ (Poisson), $(1-p+p{e}^t{)}^n$ (binomial). Pattern-matching is faster than differentiating. "MGFs multiply for sums." Multiply, recognize, read off parameters. This is the single fastest technique for distribution identification on Exam P.

The full lesson (about 1,862 words, 12 min read) adds 2 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 2a
• 2b
• 2c
• 2d
• 2e
• 2f

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