Homework 7 - them there are more than 16.3 ounces. What are...

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550.420 Introduction to Probability Fall 2011 Homework 7 Due Thursday October 20 1. (2 points) Let X be a standard Cauchy random variable. Show that E [ | X | α ] exists if 0 < α < 1 and does not exist if α 1 . 2. (2 points) A farmer Phil Dert who has two boards of lumber of lengths a and b (with a < b ) decides to build a pen in the shape of a triangle for his chickens. He sends his idiot son out to cut the longer piece, and the boy, without any thought as to the ultimate purpose, makes a cut on the board of length b at a randomly selected point. What are the chances that the resulting pieces and the piece of length a can be used to form a triangular pen? 3. (2 points) The amount of soft drink Sargent Pepper in a bottle is a random variable. Suppose that in 7% of the bottles containing this soft drink there are less than 15.5 ounces, and in 10% of
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Unformatted text preview: them there are more than 16.3 ounces. What are the mean and standard deviation of the amount of soft drink in a randomly selected bottle? 4. (2 points) In a Lone Rock, Iowa, the length of residence of a family in a home is normal with mean 80 months and variance 900. (Negative times are allowed, because they have secretly invented a time machine.) What is the probability that of 12 independent families, living on a certain street of the town, at least three will have lived there more than eight years? 5. (2 points) The random variable X is said to have the double exponential distribution if its density function is f X ( x ) = ce-| x | . (a) Find the value of c that makes this a density. (b) Prove that E [ X 2 n ] = (2 n )! and E [ X 2 n +1 ] = 0 for all n = 1 , 2 , 3 , . . . . 1...
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