Conditional Density Function

Suppose that a random variable X has density /(x) and that [a, b] is a certain closed interval such that P(a < X < b) > 0. Then, for any closed interval [xb x2] contained in [a, b], we have from Theorem 2.4.1

. P(xi < X < X2)

(3.3.2) P(x і <X<x2a<X<6) = ———————————————– •

P{a<X<b)

Now we want to ask the question: Is there a function such that its integral over [xb x2] is equal to the conditional probability given in (3.3.2)? The answer is yes, and the details are provided by Definition 3.3.2.

definition 3.3.2 Let X have density /(x). The conditional density of X given a ^ X ^ b, denoted by /(x | a ^ X < b), is defined by

(3.3.3) /(x I a < X ^ b) = — for a < x < b,

f(x)dx

J a

= 0 otherwise,

provided that Jbaf(x)dx Ф 0.

We can easily verify that /(x | a ^ X ^ b) defined above satisfies

(3.3.4) P(xi <X^x2|a^X^6)= J 2 f{x a ^ X ^ b)dx

J*1

whenever a, b D [x1;x2], as desired. From the above result it is not difficult to understand the following generalization of Definition 3.3.2.

DEFINITION 3.3.3 Let X have the density f(x) and let S be a subset of the real line such that P(X Є 5) > 0. Then the conditional density of X given X Є S, denoted by /(x | S), is defined by

, /(*)

(3.3.5) f(x S) =—————— for x Є S,

P(X Є S) = 0