Week 6

This week, we will be discussing MGFs.
Published

September 29, 2025

Learning Outcomes

First Lecture

  • Moment Generating Functions

  • MGF Properties

Second Lecture

  • R Lab Continuous Random Variables

Lecture

Tuesday Slides | Thursday Slides

Videos

Section Tuesday Thursday
001 Video Video
002 Video Video

Important Concepts

First Lecture

Moments

The \(k\)th moment is defined as the expectation of the random variable, raised to the \(k\)th power, defined as \(E(X^k)\).

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:

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

MGF Properties

Linearity

Let \(X\) follow a distribution \(f\), with the an MGF \(M_X(t)\), the MGF of \(Y=aX+b\) is given as

\[ M_Y(t) = e^{tb}M_X(at) \]

Let \(X\) and \(Y\) be two random variables with MGFs \(M_X(t)\) and \(M_Y(t)\), respectively, and are independent. The MGF of \(U=X-Y\)

\[ M_U(t) = M_X(t)M_Y(-t) \]

Uniqueness

Let \(X\) and \(Y\) have the following distributions \(F_X(x)\) and \(F_Y(y)\) and MGFs \(M_X(t)\) and \(M_Y(t)\), respectively. \(X\) and \(Y\) have the same distribution \(F_X(x)=F_Y(y)\) if and only if \(M_X(t)=M_Y(t)\).