35. If two loads are applied to a cantilever beam as shown in the accompanying drawing, the bending moment at 0 due to the loads is a1X1  a2X2.

a. Suppose that X1 and X2 are independent rv s with means 2 and 4 kips, respectively, and standard deviations .5 and 1.0 kip, respectively. If a15 ft and a2  10 ft, what is the expected bending moment and what is the standard deviation of the bending moment?

b. If X1 and X2 are normally distributed, what is the probability that the bending moment will exceed 75 kip-ft?

c. Suppose the positions of the two loads are random variables. Denoting them by A1 and A2, assume that these variables have means of 5 and 10 ft, respectively, that each has a standard deviation of .5, and that all Ai s and Xi s are independent of one another. What is the expected moment now?

d. For the situation of part (c), what is the variance of the bending moment?

e. If the situation is as described in part

(a) except that Corr(X1, X2)  .5 (so that the two loads are not independent), what is the variance of the bending moment?