Suppose that when the pH of a certain chemical compound is 5.00, the pH measured by a randomly selected beginning chemistry student is a random variable with mean 5.00 and standard deviation .2. A large batch of the compound is subdivided and a sample given to each student in a morning lab and each student in an afternoon lab. Let the average pH as determined by the morning students and the average pH as determined by the afternoon students.

a. If pH is a normal variable and there are 25 students in each lab, compute P(.1  .1). [Hint:  is a linear combination of normal variables, so is normally distributed. Compute and .] sX2Y mX2Y X Y X Y Y

X

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 a1 5 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?