Agenda

<aside> 📌 Reminders

• Quiz next week </aside>

Moment Generating Functions (MGFs)

Introduction

Moment Generating Functions (MGFs) provide a way to summarize all the moments of a random variable in a single function.

Definition

The MGF of a random variable $X$ is defined as:

$$ m_X(t) = \mathbb{E}[e^{tX}] $$

where $t$ is a real number for which the expectation exists.

Key Properties

1. Existence: The MGF exists if $m_X(t)$ is finite for $t$ in some interval around 0.
2. Moments from MGF: The $n^{\text{th}}$ moment of $X$ (i.e., $\mathbb{E}[X^n]$) can be found by differentiating the MGF $n$ times and evaluating at $t = 0$:

$$ \mathbb{E}[X^n] = m_X^{(n)}(0) $$

where $m_X^{(n)}(t)$ denotes the $n^{\text{th}}$ derivative of $m_X(t)$.

Applications

• Finding Mean and Variance: For the mean (\(\mu\)) and variance (\(\sigma^2\)):
  • Mean: $\mu = m_X'(0)$
  • Variance: $\sigma^2 = m_X''(0) - (m_X'(0))^2$
• Determining Distribution: If two random variables have the same MGF, they have the same distribution (under certain conditions).
• Sum of IID RVs: If $Y = \sum^n_i X_i$, then $m_Y(t) = \Pi_i^n m_{X_i}(t)$

MGFs of common distributions

Practice