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 aX+aX. (0, 1)- (x, 1-x)

a. Suppose that X, and X, are independent rv's with means 2 and 4 kips, respectively, and standard deviations .5 and 1.0 kip, respectively. If

a, 5 ft and a = 10 ft, what is the expected bending moment and what is the standard deviation of the bending moment?

b. If X, and X, are normally distributed, what is the proba- bility that the bending moment will exceed 75 kip-ft?

c. Suppose the positions of the two loads are random vari- ables. Denoting them by A, 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 4,'s and X'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 inde- pendent), what is the variance of the bending moment?