Higher Order Moments
Moments describe different aspects of a distribution's shape. The \(n\)th central moment is: $$ \mu_n = E[(X - E[X])^n] $$
Central Moments
- μ₀: Always 1
- μ₁: Always 0
- μ₂: Variance (spread)
- μ₃: Skewness (asymmetry)
- μ₄: Kurtosis (tailedness)
Raw Moments
About the origin:
$$ m_n = E[X^n] $$
- m₁: Mean
- m₂: E[X²]
How Moments Affect Shape
Interactive: Distribution Shape Explorer
Mean
5.0
Variance
2.5
Skewness
0.0
Moment Generating Function (MGF)
Definition
$$ M_X(t) = E[e^{tX}] = \sum_x e^{tx} p(x) $$
Key Property: The \(n\)th derivative of the MGF at \(t=0\) gives the \(n\)th raw moment: $$ E[X^n] = M_X^{(n)}(0) $$
Understanding MGF: Taylor Series
The MGF works because \(e^{tX}\) can be expanded as a Taylor series:
$$ e^{tX} = $$
$$ 1 $$
$$ + tX $$
$$ + \frac{(tX)^2}{2!} $$
$$ + \frac{(tX)^3}{3!} $$
$$ + \cdots $$
MGF Explorer
Select a Distribution
MGF:
$$ M_X(t) = (0.5e^t + 0.5)^{10} $$
E[X] = M'(0)
5.0
E[X²] = M''(0)
27.5
Var(X)
2.5