1. (Adapted from Larson, Intro. to Probability Theory, 1969) A nursery specializes in the installation of circular flower beds. When laying out the circle, a workman puts a peg in the center and cuts a length of rope (already tied in a loose loop to the stake) equal to the radius of the desired circle, and uses this to mark out the bed on the ground. Assume the desired radius is r meters. Also assume the workman is a little sloppy, and is equally likely to cut the rope to any length within the interval (r - 0.1, r + 0.1). Let X be the length of the rope from stake to cut end.
a. Determine the probability density function for X.
b. Let Y be the surface area of the circle. State Y in terms of X.
c. Determine the probability density function for Y.
d. Find E(Y)
e. Find P(Y > r2).
2. The Weibull density function is given by
f(y) = (1/) m ym-1 exp(-ym /) if 0< y <; and f(y) = 0 otherwise,
where  and m are positive constants. This density function is often used as a model for the length of life of physical systems. Suppose Y has the Weibull density just given.
a. Find the density function of U = Ym.
b. Find E(Yk) for any positive integer k.
c. Suppose W has an exponential distribution with mean . Prove that Y = W has a Weibull density with  =  and m = 2.
d. Find E(Wk/2)
3. A small orchard contains 5 Winesap apple trees and 3 Mutsu apple trees, all at peak bearing age. The yearly harvests for Winesap apple trees at that stage are normally distributed with mean 1 and variance 2, while the yearly harvest for Mutsu apple trees at that stage are normally distributed with mean 2 and variance 22.
a. What is the expected yearly harvest for this orchard?
b. Assume you have lists of yearly harvest amounts per tree and you can randomly select n of the Winesap harvest amounts and m of the Mutsu harvest amounts. Suppose 2 is known. Construct a 95% confidence interval for the expected yearly harvest for this orchard.
c. Under the same conditions as in part b, but with 2 unknown, construct a 95% confidence interval for the expected yearly harvest for this orchard.
The solution contains the determination of probability density functions of various transformations of random variable. The identification of probability density function for a real-world problem is also discussed in the solution.