Marginal Distributions

Discrete

Marginal Distributions (marginal pdfs) of $X$ and $Y$ :

where $f (x, y)$ is their joint pmf

box with 9 balls, 2 white, 4 black (Y), 3 red (X)

$P (X = x, Y = y)$	x = 0	x = 1	x = 2	$f_{y} (y) = P (Y = y)$
y = 0	1/36	6/36	3/36	10/36
y = 1	8/36	12/36	0	20/36
y = 2	6/36	0	0	6/36
$f_{x} = P (X = x)$	15/36	18/36	3/36

If X and Y are continuous random variables and $f (x, y)$ is the value of their joint pdf at $(x, y)$ , the functions given by

\begin{aligned} f_{X} (x) = \int_{- \infty}^{\infty} f_{X, Y} (x, y) d y a n d f_{Y} (y) = \int_{- \infty}^{\infty} f_{X, Y} (x, y) d x \end{aligned}

are called the marginal distributions (marginal pdfs) of X and Y respectively.