h8 - Statistics 5101 Fall 2010 Homework Assignment 8 Solve...

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Statistics 5101, Fall 2010: Homework Assignment 8 Solve each problem. Explain your reasoning. No credit for answers with no explanation. If the problem is a proof, then you need words as well as formulas. Explain why your formulas follow one from another. 8-1. Let X have the standard Cauchy distribution, which has PDF defined in the brand name distributions handout f ( x ) = 1 π · 1 1 + x 2 , -∞ < x < . (a) Find the quantile function for X . (b) Find the median of X . (c) Find the lower and upper quartiles of X . Hint: the indefinite integral of 1 / (1 + x 2 ) is arc tangent of x (inverse of the tangent function). 8-2. Suppose X has the Exp(1) distribution. (a) What is the best prediction of the value of X if minimizing expected squared error is the criterion? (b) What is the best prediction of the value of X if minimizing expected absolute error is the criterion? In this problem, we want numeric answers so we can see how different they are. 8-3. Suppose X has the Gam(2 , 1) distribution.
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