## 2.10.2 ExpectationGiven a continuous random variable X with pdf f _{x}(x), the expectation
or expected value of the function g(X) isAll the results concerning expected values derived in Section 2.7 carry over in the obvious way, with summations replaced by integrations. Exercise 2.13: Expected Values, Revisited Verify that the results of
Exercises 2.6-2.9 also apply to continuously distributed random variables.Example 1: (continued)Here we continue our triangle problem initially described in Example 1, Section 2.1. We restate the problem as follows: "Two points X _{1} and X_{2}
are marked randomly and independently on a stick of length 1 meter."a. Determine the probability that a triangle can be formed with the three pieces obtained by cutting the stick at the marked points. b. Determine the conditional pdf for X _{1}, given that a triangle can be
formed.C. Determine the conditional pdf for X _{2}, given that
X_{1} = 1/4 and a tri-angle cannot be formed.Solution:a. First we must interpret the word "random." In the absence of any further information, the most reasonable interpretation is that X _{1} and
X_{2} are uniformly and independently distributed over [0, 1]. Thus, the
joint pdf for (X_{1}, X_{2}) isLetting be the event that a triangle can be formed, we recall that corresponds to the two triangular regions of the (X _{1}, X_{2}) sample space
shown in Figure 2.1. Since the area of each is 1/8 and since the joint
pdf is uniform with height 1, we obtain by inspection that P{} = 1/4. If the pdf were not uniform, we would have to evaluate the following
integral:as we obtained by inspection. b. Given that a triangle can be formed, the conditional (X _{1}, X_{2}) sample
space comprises the two triangular regions over which we have just
integrated. If one invokes the definition of the marginal pdf in terms
of probabilities of lying within infinitesimal strips,then one can see from Figure 2.1 that this "strip probability" increases linearly from 0 to a maximum at X _{1} = 1/2 and then decreases
linearly (and symmetrically) back to zero. Thus, by inspection,as anticipated. c. The conditioning information is that X _{1} = 1/4 and a triangle cannot be
formed. By inspection of Figure 2.1, the conditional pdf for X_{2} isThis pdf, which is displayed in Figure 2.10, is derived by considering a strip of infinitesimal width dx _{1} at x_{1} = 1/4. Integration requires that c' = 4/3.We could continue the process of conditioning indefinitely and, in theory, incur no additional problems. For instance, let A = event | X _{2} - 1/2 | > 1/8; this means that X_{2} is either less than 3/8 or greater
than 5/8. Then |