Stochastic

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Unformatted text preview: on (a,b). Suppose X has density function fX (x) = 1/(b a) for a < x < b and 0 otherwise. In this case EX = Z b a x b a dx = b2 a2 (b + a) = 2(b a) 2 since b2 a2 = (b a)(b + a). Notice that (b + a)/2 is the midpoint of the interval and hence the natural choice for the average value of X . A little more calculus gives EX = 2 Z b a x2 b a dx = b3 a3 b2 + ba + a2 = 3(b a) 3 since b3 a3 = (b a)(b2 + ba + a2 ). Squaring our formula for EX gives (EX )2 = (b2 + 2ab + a2 )/4, so var (X ) = (b2 2ab + a2 )/12 = (b a)2 /12 To help explain the answers we have found in the last two example we use Theorem A.3. If c is a real number, then (a) E (X + c) = EX + c (b) var (X + c) = var (X ) (c) E (cX ) = cEX (d) var (cX ) = c2 var (X ) Uniform distribution on (a,b). If X is uniform on [(a b)/2, (b a)/2] then EX = 0 by symmetry. If c = (a + b)/2, then Y = X + c is uniform on [a, b], so it follows from (a) and (b) of Theorem A.3 that EY = EX + c = (a + b)/2 var (Y ) = var (X ) From the second formula we se...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell University (Engineering School).

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