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: 7


$p(x)$ is a probability density function on $[1, \infty)$. and $\mu=\dfrac{3}{2} $

Work Step by Step

We have: $\int_{1}^{\infty} 3x^{-4} \ dx=\lim\limits_{a \to \infty}\int_{1}^{a} 3x^{-4} \ dx\\=(-1) \lim\limits_{a \to \infty} \dfrac{1}{a^3}-1\\=(-1) (-1)\\=1 $ This implies that $p(x)$ is a probability density function on $[1, \infty)$. The mean value is: $\mu=\int_{1}^{\infty} (x) 3x^{-4} \ dx \\=\lim\limits_{a \to \infty}\int_{1}^{a} 3x^{-3} \ dx\\=\lim\limits_{a \to \infty} [\dfrac{-3}{2}x^{-2}]_1^a\\=\dfrac{-3}{2} \lim\limits_{a \to \infty} \dfrac{1-a^2}{a^2} \\=\dfrac{3}{2} $
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