Marginal and Conditional Distributions for Actuarial Exam P

Marginal Distributions

The marginal PDF of X is obtained by integrating out Y: f_X(x) = integral of f(x, y) dy over all y. Similarly, f_Y(y) = integral of f(x, y) dx. For discrete variables, sum instead of integrate. The marginal distribution captures the behavior of one variable ignoring the other.

For example, if f(x, y) = 6(1 - y) for 0 < x < y < 1, then f_X(x) = integral from x to 1 of 6(1 - y) dy = 6[(y - y^2/2)] evaluated from x to 1 = 6[(1 - 1/2) - (x - x^2/2)] = 3(1 - x)^2 for 0 < x < 1.

Conditional Distributions

The conditional PDF of Y given X = x is f(y|x) = f(x, y) / f_X(x). This is valid whenever f_X(x) > 0. The conditional distribution acts like a regular probability distribution in y for each fixed x. From it you can compute conditional expectations E[Y|X = x] = integral of y * f(y|x) dy.

Continuing the example, f(y|x) = 6(1 - y) / (3(1 - x)^2) = 2(1 - y) / (1 - x)^2 for x < y < 1. The conditional mean E[Y|X = x] can be found by integrating y * f(y|x) from x to 1.

Applications on Exam P

A frequent Exam P pattern gives the conditional distribution of X given Theta and the marginal distribution of Theta, then asks for the marginal distribution of X or E[X]. Use the law of total expectation: E[X] = E[E[X|Theta]]. This is the "double expectation" formula. For variance, the total variance formula applies: Var(X) = E[Var(X|Theta)] + Var(E[X|Theta]). Both formulas are essential for Exam P.