Moment Generating Function — Definition, Formula & Examples

A moment generating function (MGF) is a function that encodes all the moments (mean, variance, and higher) of a random variable into a single expression. You obtain the

n

th moment by taking the

n

th derivative of the MGF and evaluating it at zero.

For a random variable $X$ , the moment generating function is defined as $M_X(t) = E[e^{tX}]$ , provided this expectation exists for $t$ in some open interval containing zero. When it exists, $M_X(t)$ uniquely determines the probability distribution of $X$ , and the $n$ th moment is given by $E[X^n] = M_X^{(n)}(0)$ .

Key Formula

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

Where:

$M_X(t)$ = Moment generating function of random variable X, as a function of t
$E$ = Expected value operator
$t$ = Real-valued parameter near zero
$X$ = Random variable

How It Works

To use an MGF, first compute

E[e^{tX}]

using the distribution's PMF or PDF. Once you have a closed-form expression for

M_X(t)

, differentiate it

n

times with respect to

t

and set

t = 0

to extract the

n

th moment. The first derivative at zero gives the mean

E[X]

, and combining the first two derivatives gives the variance via

\text{Var}(X) = M_X''(0) - [M_X'(0)]^2

. MGFs are also powerful for proving that sums of independent random variables follow known distributions, since

M_{X+Y}(t) = M_X(t) \cdot M_Y(t)

when

X

and

Y

are independent.

Worked Example

Problem: Find the MGF of an exponential random variable with rate λ = 3, then use it to find the mean.

Step 1: Write the MGF using the PDF

f(x) = 3e^{-3x}

for

x \geq 0

M_X(t) = \int_0^\infty e^{tx} \cdot 3e^{-3x}\,dx = 3\int_0^\infty e^{-(3-t)x}\,dx

Step 2: Evaluate the integral (valid for

t < 3

M_X(t) = \frac{3}{3 - t}

Step 3: Differentiate and evaluate at

t = 0

to find the mean.

M_X'(t) = \frac{3}{(3-t)^2} \implies M_X'(0) = \frac{3}{9} = \frac{1}{3}

Answer: The MGF is

M_X(t) = \frac{3}{3 - t}

for

t < 3

, and the mean is

E[X] = \frac{1}{3}

Why It Matters

MGFs appear throughout mathematical statistics whenever you need to derive the distribution of a sum of independent random variables or prove convergence results like the Central Limit Theorem. In actuarial science and reliability engineering, MGFs provide a compact way to characterize risk distributions and compute tail probabilities.

Common Mistakes

Mistake: Forgetting that the MGF does not exist for all distributions (e.g., the Cauchy distribution has no MGF).

Correction: Always verify that

E[e^{tX}]

is finite in an open interval around

t = 0

before using the MGF. If it does not exist, use the characteristic function instead.