Functions of a Random Variable

What

We define a new random variable \(Y = g(X)\), where \(X\) is a known random variable and \(g\) is a deterministic function. This allows us to study the distribution of transformed quantities, like kinetic energy from velocity, Celsius from Fahrenheit, or squared values from a normal distribution.

Why

Many real-world quantities are functions of more basic random variables.
Enables modeling derived or transformed metrics, like energy, signal power, or log-returns.
Critical in probability and statistics for creating new distributions (e.g., chi-square from normal).

How

To find the distribution of \(Y = g(X)\):

Do not directly plug \(g(X)\) into the PDF of \(X\) → this is incorrect.
Instead, follow this valid procedure:

Why you cannot directly plug \(g(X)\) into the PDF\(f_X(x)\):

The PDF \(f_X(x)\) gives the probability density at a particular Celsius temperature \(x\).
\(g(X)\) gives the Fahrenheit temperature corresponding to Celsius temperature \(X\).
To find the PDF of \(Y\), you cannot just substitute \(g(x)\) inside \(f_X\) like \(f_X(g(x))\). That won't give you a proper PDF of \(Y\).

1. Use the CDF:

\[ F_Y(y) = P(Y \leq y) = P(g(X) \leq y) \]

Transform this into an expression involving \(X\) and use \(F_X\) (the known CDF of \(X\)).

2. Differentiate the CDF to get the PDF:

You cannot go PDF → PDF directly. You must pass through the CDF.

\[ f_Y(y) = \frac{d}{dy} F_Y(y) \]

This gives a well-defined PDF for \(Y\), which reflects the correct transformation.

Example 1: Linear Transformation

Let \(X \sim \mathcal{N}(\mu, \sigma^2)\) and define \(Y = aX + b\) Then to get the CDF:

\[ \begin{align*} F_Y(y) = P(Y \lt y) = P(aX + b < y) \\ =P(X < \frac{y-b}{a}) \\ =F_X\left(\frac{y - b}{a}\right) \end{align*} \]

Differentiating CDF to get the PDF:

\[ \begin{align*} f_Y(y) = \frac{d}{dy}F_X\left(\frac{y - b}{a}\right) \\ =\frac{1}{a} f_X\left(\frac{y - b}{a}\right) \end{align*} \]

Result: \(Y \sim \mathcal{N}(a\mu + b, a^2\sigma^2)\)