If the event of interest only involves one of the two variables, the double-integral simplifies considerably. For example,

\[ P(a \lt X \lt b) \;=\; \int_a^b \int_{-\infty}^{\infty} f(x,y) \;dy \; dx \;=\; \int_a^b f_X(x) \; dx \]

where

\[ f_X(x) \;=\; \int_{-\infty}^{\infty} f(x,y) \;dy \]

The function \(f_X(x)\) is called the marginal probability density function of \(X\). The marginal probability density function of \(Y\) is similarly defined as

\[ f_Y(y) \;=\; \int_{-\infty}^{\infty} f(x,y) \;dx \]

Marginal distributions

A marginal pdf gives the distributions of one variable if nothing is known about the other variable.

Marginal pdfs can be found by integration, but geometry can occasionally be used — \(f_X(x)\) is the area of a cross-section through the joint pdf at \(x\).

Example

The random variables \(X\) and \(Y\) have joint probability density function

\[ f(x,y) \;=\; \begin{cases} x+y & \quad\text{if }0 \lt x \lt 1 \text{ and }0 \lt y \lt 1 \\ 0 & \quad\text{otherwise} \end{cases} \]

What is the marginal pdf of \(X\)?

(Solved in full version)