Agenda

<aside> 📌 Reminders

• Final exam December 11, 2024 </aside>

Change of Variable Formula for Densities

Recall the change of variable formula from last week (where $X$ is a continuous random variable and $g$ is a monotonic function,

$$ f_Y(y) = f_X\big(g^{-1}(y)\big) \cdot \left| \frac{d}{dy} g^{-1}(y) \right|,  $$

Example

If $X \sim \text{Unif}(0,1)$ (PDF $f_X(x) = 1$ for $0 \leq x \leq 1$) and $Y = X^2$:

• $g(x) = x^2$, so $g^{-1}(y) = \sqrt{y}$.
• Derivative: $\frac{d}{dy} g^{-1}(y) = \frac{1}{2\sqrt{y}}$.
• PDF: $f_Y(y) = f_X(\sqrt{y}) \cdot \left| \frac{1}{2\sqrt{y}} \right| = 1 \cdot \frac{1}{2\sqrt{y}} = \frac{1}{2\sqrt{y}}$ for $0 \leq y \leq 1$.

Moment Generating Function (MGF)

Defintion

The MGF of a random variable $X$ is:

$$ M_X(t) = \mathbb{E}[e^{tX}] = \int_{-\infty}^\infty e^{tx} f_X(x) \, dx.  $$

Purpose

• Moments: Derivatives of the MGF give moments of $X$. Specifically, the $n$-th moment is:

$$ \mathbb{E}[X^n] = M_X^{(n)}(0) = \left. \frac{d^n}{dt^n} M_X(t) \right|_{t=0}.  $$

• Uniqueness: If the MGF exists in a neighbourhood of $t = 0$, it uniquely determines the distribution of $X$.

Common MGFs

• Normal $\mathcal{N}(\mu, \sigma^2)$:

  $$ M_X(t) = \exp\left(\mu t + \frac{\sigma^2 t^2}{2}\right).  $$