Moments and MGFs

Moments are summary statistics of a distribution. They capture location, spread, and shape.

Moments (raw vs central)

Raw moment: $E[X^n]$
Central moment: $E[(X-\mu)^n]$ where $\mu = E[X]$

Common moments:

Mean: $E[X]$
Variance: $\mathrm{Var}(X)=E[(X-\mu)^2] = E[X^2]-\mu^2$
Skewness (shape): based on $E[(X-\mu)^3]$
Kurtosis (tails/peakedness): based on $E[(X-\mu)^4]$

Moment Generating Function (MGF)

The MGF is

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

Why it’s useful:

Differentiating at $t=0$ gives raw moments:

M_X^{(n)}(0)=E[X^n].

If an MGF exists in a neighborhood around $0$ , it uniquely identifies the distribution.

Note: Some distributions don’t have an MGF (e.g., Cauchy). In those cases people often use the characteristic function instead.

Test Your Knowledge

Example: Using the MGF for a Poisson

The Moment Generating Function of a Poisson distribution with rate $\lambda$ is $M_X(t) = e^{\lambda(e^t - 1)}$ . Use the MGF to find the First Moment (the Mean) of the Poisson distribution.

View Step-by-Step Solution

To find the first moment, we take the first derivative of the MGF with respect to $t$ , and evaluate it at $t=0$ .

$\frac{d}{dt} M_X(t) = \frac{d}{dt} e^{\lambda(e^t - 1)}$

Using the chain rule:

$M_X'(t) = e^{\lambda(e^t - 1)} \cdot \frac{d}{dt}[\lambda(e^t - 1)] = e^{\lambda(e^t - 1)} \cdot (\lambda e^t)$

Evaluate at $t=0$ :

$M_X'(0) = e^{\lambda(e^0 - 1)} \cdot (\lambda e^0) = e^{\lambda(1 - 1)} \cdot \lambda = \lambda$

Thus, the Mean of the Poisson distribution is exactly $\lambda$ .

Gamma & Beta Transformations