Agenda

<aside> 📌 Reminders

Change of Variable Formula for Multiple Integrals

When we transform variables in multiple integrals, we must adjust for changes in the "volume element" of the integral. This adjustment is made using the Jacobian determinant.

Key Steps:

  1. Identify the transformation $\mathbf{u} = g(\mathbf{x})$ that maps each point $\mathbf{x}$ in the original variables to a new point $\mathbf{u}$.
  2. Compute the Jacobian matrix $J$ of partial derivatives for this transformation.
  3. The Jacobian determinant, $| \det(J) |$, scales the volume element to account for stretching or compressing of the space due to the transformation.

Example:

For a transformation from $(x, y)$ to polar coordinates $(r, \theta)$, the Jacobian determinant $| \det(J) | = r$ allows us to write:

$$ \int \int f(x, y) \, dx \, dy = \int \int f(r \cos \theta, r \sin \theta) \cdot r \, dr \, d\theta  $$

Univariate Transformations of Random Variables

To find the distribution of a transformed random variable $Y = g(X)$ where $X$ is continuous:

  1. Determine the range of $Y$.

  2. Find the inverse function $X = g^{-1}(Y)$ (if it exists) and compute the derivative $\frac{dX}{dY}$.

  3. Use the change of variable formula for probability density functions:

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

Example:

For $Y = X^2$ where $X \sim N(0, 1)$, we calculate $f_Y(y)$ by considering the probability of each possible $X$ value that maps to $Y = y$.

Convolution of Random Variables

The convolution of two independent random variables $X$ and $Y$, resulting in a sum $Z = X + Y$, helps us find the distribution of $Z$.

Convolution Formula: