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Mathematical Statistics

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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 > &#61552;r2).

2. The Weibull density function is given by
f(y) = (1/&#61537;) m ym-1 exp(-ym /&#61537;) if 0< y <&#61605;; and f(y) = 0 otherwise,
where &#61537; 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 &#61538;. Prove that Y = &#61654;W has a Weibull density with &#61537; = &#61538; 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 &#61549;1 and variance &#61555;2, while the yearly harvest for Mutsu apple trees at that stage are normally distributed with mean &#61549;2 and variance 2&#61555;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 &#61555;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 &#61555;2 unknown, construct a 95% confidence interval for the expected yearly harvest for this orchard.