## Definition A ==Marginal Probability Function== gives the distribution of a single random variable obtained by summing (discrete) or integrating (continuous) the joint probability function over all values of the other variable(s). For continuous random variables: $ f_X(x) = \int_{-\infty}^{\infty} f(x, y)\, dy $ > [!example]- If $f(x, y) = 24xy$ for $0 \leq x \leq 1$, $0 \leq y \leq 1$, and $x + y \leq 1$, what is $f_X(x)$? > $ f_X(x) = \int_0^{1-x} 24xy\, dy = 24x \cdot \frac{(1-x)^2}{2} = 12x(1-x)^2, \quad 0 \leq x \leq 1 $