If two loads are applied to a cantilever beam as shown in the accompanying drawi

Question

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 a_1X_1 + a_2X_2. Suppose that X_1 and X_2 are independent rv's with means 4 and 8 kips, respectively, and standard deviations 1.4 and 2.8 kip, respectively. If a_1 = 9 ft and a_2 = 18 ft, 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 If X_1 and X_2 are normally distributed, what is the probability that the bending moment will exceed 75 kip-ft? (Round your answer to four decimal places.) Suppose the positions of the two loads are random variables. Denoting them by A_1 and A_2, assume that these variables have means of 9 and 18 ft, 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 For the situation of part (c), what is the variance of the bending moment? () kip-ft^2 If the situation is as described in part (a) except that Corr(X_1 X_2) = 0.5 (so that the two loads are not independent), what is the variance of the bending moment? kip-ft^2

Explanation / Answer

(d) 163.25

This result is obtained by using the formula for variance of product of two independant random variables X and Y :

Var(XY)= E(X2Y2)(E(XY))2 = Var(X)Var(Y) + Var(X)(E(Y))2 + Var(Y)(E(X))2.

(e) 2860.92

This result is obtained by using the formula for variance of linear function of two correlated random variables X and Y :

Var(aX+bY)= a2*Var(X) + b2*Var(Y) + 2ab*Cov(X,Y)

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