Week 4

This week, we will discuss Expected Values and Moment Generating Function.
Published

September 12, 2025

Learning Outcomes

First Lecture

  • Moment Generating Functions

  • Expected Value Properties

Second Lecture

  • Expected Values
  • MGF

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

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

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

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