DSpace at VNU: Continuous Distributions: Probability Examples c-6 - eBooks and textbooks from bookboon.com tài liệu, giá...
Continuous Distributions Probability Examples c-6 Leif Mejlbro Download free books at Leif Mejlbro Probability Examples c-6 Continuous Distributions Download free eBooks at bookboon.com Probability Examples c-6 – Continuous Distributions © 2009 Leif Mejlbro & Ventus Publishing ApS ISBN 978-87-7681-522-6 Download free eBooks at bookboon.com Continuous Distributions Contents Contents Introduction 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 1.11 1.12 Some theoretical background The exponential distribution The normal distribution 2-dimensional normal distributions Conditional normal distribution Sums of independent normal distributed random variables The Central Limit Theorem The Maxwell distribution The Gamma distribution The distribution The t distribution The F distribution Estimation of parameters 7 10 11 11 13 13 14 15 17 17 The Exponential Distribution 20 The Normal Distribution 31 The Central Limit Theorem 46 The Maxwell distribution 80 The Gamma distribution 83 The normal distribution and the Gamma distribution 117 Convergence in distribution 122 www.sylvania.com We not reinvent the wheel we reinvent light Fascinating lighting offers an ininite spectrum of possibilities: Innovative technologies and new markets provide both opportunities and challenges An environment in which your expertise is in high demand Enjoy the supportive working atmosphere within our global group and beneit from international career paths Implement sustainable ideas in close cooperation with other specialists and contribute to inluencing our future Come and join us in reinventing light every day Light is OSRAM Click on the ad to read more Download free eBooks at bookboon.com Continuous Distributions CHAPTER The distribution 126 10 The F distribution 127 11 The F distribution and the t distribution 130 12 Estimation of parameters 131 Index 167 360° thinking Discover the truth at www.deloitte.ca/careers © Deloitte & Touche LLP and affiliated entities Click on the ad to read more Download free eBooks at bookboon.com Introduction Continuous Distributions Introduction This is the sixth book of examples from the Theory of Probability This topic is not my favourite, however, thanks to my former colleague, Ole Jørsboe, I somehow managed to get an idea of what it is all about The way I have treated the topic will often diverge from the more professional treatment On the other hand, it will probably also be closer to the way of thinking which is more common among many readers, because I also had to start from scratch The prerequisites for the topics can e.g be found in the Ventus: Calculus series, so I shall refer the reader to these books, concerning e.g plane integrals Unfortunately errors cannot be avoided in a first edition of a work of this type However, the author has tried to put them on a minimum, hoping that the reader will meet with sympathy the errors which occur in the text Leif Mejlbro 27th October 2009 Download free eBooks at bookboon.com Some theoretical background Continuous Distributions Some theoretical background 1.1 The exponential distribution A random variable X follows an exponential distribution with parameter a > 0, if its distribution function F (x) is given by ⎧ for x ≥ 0, ⎨ − e−ax , F (x) = ⎩ 0, for x < The corresponding frequency f (x) is given by ⎧ for x ≥ 0, ⎨ a e−ax , f (x) = ⎩ 0, for x < We have for an exponentially distributed random variable X with parameter a > 0, E{X} = a and V {X} = a2 In general, if X is exponentially distributed, then P {X > s + t | X > s} = P {X > t}, for s, t > 0, which is equivalent with the formula P {X > s + t} = P {X > s} · P {X > t}, for s, t > We say that the exponential distribution is forgetful In practice, the exponential distribution often occurs as a distribution of lifetimes, which is in particular the case in queuing theory In this case the forgetfulness is of paramount importance An exponentially distributed random variable X with parameter a > is a special gamma distribution (cf the following), so one also writes, X ∈ Γ 1, a for the exponential distribution Another type of generalized exponential distributions is the Weibull distribution with parameters a, b > This is given by the distribution function ⎧ for x ≥ 0, ⎨ − exp −a xb , F (x) = ⎩ 0, for x < We note that we get the exponential distribution for b = The Weibull distribution is used in connection with the theory of reliability Download free eBooks at bookboon.com ...Leif Mejlbro Probability Examples c-6 Continuous Distributions Download free eBooks at bookboon.com Probability Examples c-6 – Continuous Distributions © 2009 Leif Mejlbro... Continuous Distributions © 2009 Leif Mejlbro & Ventus Publishing ApS ISBN 97 8-8 7-7 68 1-5 2 2-6 Download free eBooks at bookboon.com Continuous Distributions Contents Contents Introduction 1.1 1.2 1.3 1.4... our global group and beneit from international career paths Implement sustainable ideas in close cooperation with other specialists and contribute to inluencing our future Come and join us in reinventing