Moment Generating Functions

Learning Outcomes

  • Moments

  • Moment Generating Functions

  • Properties

Moments and Moment Generating Functions

Moments

The expected value is the value we expect when we randomly sample from population that follows a specific distribution. The expected value of \(Y\) is

\[ E(Y)=\sum_y yP(y) \]

where \(P(y)\) is the PMF of \(Y\).

Moment Generating Functions

The moment generating functions is used to obtain the \(k\)th moment. The mgf is defined as

\[ m(c) = E(e^{tX}) \]

The \(k\)th moment can be obtained by taking the \(k\)th derivative of the mgf, with respect to \(c\), and setting \(c\) equal to 0:

\[ E(X^k)=\frac{d^km(c)}{dc}\Bigg|_{c=0} \]

Binomial Distribution

MGF

\(E(Y^2)\)

Poisson Distribution

MGF

\(E(Y^3)\)

Geometric Distribution

MGF

\(E(Y)\)

Expected Value Properties

Properties

  1. \(E(c)=c\), where \(c\) is constant
  2. \(E\{cg(Y)\}=cE\{g(Y)\}\)
  3. \(E\{g_1(Y)+\cdots+g_n(Y)\}=E\{g_1(Y)\}+\cdots+E\{g_n(Y)\}\)