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 thatX₁andX₂are independent rv's with means5and10kips, respectively, and standard deviations1.1and2.2kip, respectively. If

a₁=6ft

and

a₂=12ft,

what is the expected bending moment and what is the standard deviation of the bending moment? (Round your standard deviation to three decimal places.)

expected bending moment	kip-ft
standard deviation	kip-ft

(b) IfX₁andX₂are normally distributed, what is the probability that the bending moment will exceed 75 kip-ft? (Round your answer to four decimal places.) (c) Suppose the positions of the two loads are random variables. Denoting them byA₁andA₂, assume that these variables have means of6and12ft, respectively, that each has a standard deviation of 0.5, and that all

A_i's

and

X_i's

are independent of one another. What is the expected moment now? kip-ft (d) For the situation of part (c), what is the variance of the bending moment? (Round your answer to two decimal places.) kip-ft² (e) If the situation is as described in part (a) except that

Corr(X₁,X₂) = 0.5

(so that the two loads are not independent), what is the variance of the bending moment? kip-ft²