Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 8 - Techniques of Integration - 8.8 Probability and Integration - Exercises - Page 448: 6


$\approx 0.982$

Work Step by Step

We are given that $p(x)=Ce^{-x} e^{-e^{-x}}$ Suppose that $a =-e^{-x} \implies da=e^{-x} dx$ Consider $I=\int_{-\infty}^{\infty} Ce^{-x} e^{-e^{-x}} dx \\=C \int_{-\infty}^{\infty} e^{u} du\\=C \lim\limits_{R \to \infty}(e^{-e^{-R}}-e^{-e^{R}}) \\=C(1-0)\\=C$ We will find $P(-4 \leq X \leq 4)=\int_{-4}^{4} p(x) \ dx\\=\int_{-4}^4 e^{-x} e^{-e^{-x}} \\=[e^{-e^{-x}}]_{-4}^4 \\=e^{-e^{-4}}-e^{-e^{4}} \approx 0.982$
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