4106-08-h1 - 1 Homework Assignment 1 1. Each of the...

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1 Homework Assignment 1 1. Each of the following non-negative random variables X has a continuous distribution. You are given either the density function f ( x ) or the cumulative distribution function (cdf) F ( x ) = P ( X x ). From whichever you are given, derive the other so you have both. Then calculate the expected value E ( X ) in two ways: Directly via E ( X ) = R xf ( x ) dx and by integrating the tail E ( X ) = R P ( X > x ) dx . (a) (exponential distribution) With λ > 0 a fixed constant X has density function f ( x ) = ± λe - λx ; if x 0 0 , if x < 0. (b) (uniform distribution on ( a,b ); 0 a < b ) X has density function f ( x ) = ± 1 b - a ; if x ( a,b ) 0 , otherwise. (c) (Pareto distribution) X has cdf F ( x ) = ± 0; if x < 1 1 - 1 x 3 , if x 1 . 2. (Continued) Now repeat the above for calculating the second moment E ( X 2 ) in two ways: Directly via E ( X 2 ) = R x 2 f ( x ) dx and by use of E ( X 2 ) = Z 0 2 xP ( X > x ) dx. (1) 3. (Continued) Using your answers from above, give the variance
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4106-08-h1 - 1 Homework Assignment 1 1. Each of the...

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