Marginal Probability Function

Study guide: Exam P-1

The marginal probability function of $X$ is obtained from the Joint Probability Function $p(x,y)$ by summing over all values of $Y$ , describing the behavior of each variable in isolation.

$p_X(x) = P(X = x)$

$= \sum_y p(x, y)$

Similarly $p_Y(y) = \sum_x p(x,y)$
In the continuous case, the marginal PDF is:

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

Marginal distributions are the starting point for computing Moments for Joint Distributions and assessing independence

Finding a Marginal PMF Joint PMF: $p(1,1) = 0.2$ , $p(1,2) = 0.3$ , $p(2,1) = 0.1$ , $p(2,2) = 0.4$ . Find $p_X(x)$ .

Answer $p_X(1) = p(1,1) + p(1,2) = 0.2 + 0.3 = 0.5$ $p_X(2) = p(2,1) + p(2,2) = 0.1 + 0.4 = 0.5$