HW2 - STAT 410 Homework#2(due Friday September 12 by 3:00...

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STAT 410 Fall 2008 Homework #2 (due Friday, September 12, by 3:00 p.m.) 1. ( ~ 1.9.19 ) Let X be a nonnegative continuous random variable with p.d.f. f ( x ) and c.d.f. F ( x ) . Show that E ( X ) = ( ) ( ) ° - 0 1 x x d F . 2. Suppose that X follows a uniform distribution on the interval [ π / 2 , π / 2 ] . Find the c.d.f. and the p.d.f. of Y = tan X. 3. An insurance policy reimburses a loss up to a benefit limit of 10. The policyholder’s loss, Y , follows a distribution with density function: f ( y ) = ± ± ² ± ± ³ ´ > otherwise 0 1 if 2 3 y y What is the expected value and the variance of the benefit paid under the insurance policy?
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4. The time, T , that a manufacturing system is out of operation has cumulative distribution function F ( t ) = ± ± ² ± ± ³ ´ > µ · ¸ ¹ º - otherwise 0 2 if 2 1 2 t t The resulting cost to the company is Y = T 2 . Determine the density function of Y . 5. Suppose that X is distributed with density given by f ( x ) = ± ± ² ± ± ³ ´ > otherwise 0 1 if 2 3 x x a) Calculate the mean of X.
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