Loss Models: From Data to Decisions, 4th Edition

18.2 Conditional distributions and expectation

The formulation of the problem just presented involves the use of conditional distributions, given the risk parameter θ of the insured. Subsequent analyses of mathematical models of this nature will be seen to require a good working knowledge of conditional distributions and conditional expectation. A discussion of these topics is now presented.

Much of the material is of a review nature and, hence, may be quickly glossed over by a reader with a good background in probability. Nevertheless, there may be some material not seen before, and so this section should not be completely ignored.

Suppose that X and Y are two random variables with joint probability function (pf) or probability density function (pdf)² f_X,Y(x, y) and marginal pfs f_X(x) and f_Y(y), respectively. The conditional pf of X given that Y = y is

If X and Y are discrete random variables, then f_X|Y(x|y) is the conditional probability of the event X = x under the hypothesis that Y = y. If X and Y are continuous, then f_X|Y(x|y) may be interpreted as a definition. When X and Y are independent random variables,

and, in this case,

We observe that the conditional and marginal distributions ...