Random variables III

Random variables III tài liệu, giáo án, bài giảng , luận văn, luận án, đồ án, bài tập lớn về tất cả các lĩnh vực kinh tế...

Random variables III Probability Examples c-4 Leif Mejlbro Download free books at Leif Mejlbro Probability Examples c-4 Random variables III Download free eBooks at bookboon.com Probability Examples c-4 – Random variables III © 2009 Leif Mejlbro & Ventus Publishing ApS ISBN 978-87-7681-519-6 Download free eBooks at bookboon.com Random variables III Contents Contents Introduction Some theoretical results Maximum and minimum of random variables 20 The transformation formula and the Jacobian 34 Conditional distributions 60 Some theoretical results 72 The correlation coecient 74 Maximum and minimum of linear combinations of random variables 78 Convergence in probability and in distribution 91 Index 113 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 Introduction Random variables III Introduction This is the fourth 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 topic itself, Random Variables, is so big that I have felt it necessary to divide it into three books, of which this is the third one 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 26th October 2009 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 Some theoretical results Random variables III Some theoretical results The abstract (and precise) definition of a random variable X is that X is a real function on Ω, where the triple (Ω, F, P ) is a probability field, such that {ω ∈ Ω | X(ω) ≤ x} ∈ F for every x ∈ R This definition leads to the concept of a distribution function for the random variable X, which is the function F : R → R, which is defined by F (x) = P {X ≤ x} (= P {ω ∈ Ω | X(ω) ≤ x}), where the latter expression is the mathematically precise definition which, however, for obvious reasons everywhere in the following will be replaced by the former expression A distribution function for a random variable X has the following properties: ≤ F (x) ≤ for every x ∈ R The function F is weakly increasing, i.e F (x) ≤ F (y) for x ≤ y limx→−∞ F (x) = and limx→+∞ F (x) = The function F is continuous from the right, i.e limh→0+ F (x + h) = F (x) for every x ∈ R One may in some cases be interested in giving a crude description of the behaviour of the distribution function We define a median of a random variable X with the distribution function F (x) as a real number a = (X) ∈ R, for which P {X ≤ a} ≥ and P {X ≥ a} ≥ Expressed by means of the distribution function it follows that a ∈ R is a median, if F (a) ≥ and F (a−) = lim F (x + h) ≤ h→0− In general we define a p-quantile, p ∈ ]0, 1[, of the random variable as a number a p ∈ R, for which P {X ≤ ap } ≥ p and P {X ≥ ap } ≥ − p, which can also be expressed by F (ap ) ≥ p and F (ap −) ≤ p If the random variable X only has a finite or a countable number of values, x1 , x2 , , we call it discrete, and we say that X has a discrete distribution A very special case occurs when X only has one value In this case we say that X is causally distributed, or that X is constant Download free eBooks at bookboon.com ...Leif Mejlbro Probability Examples c-4 Random variables III Download free eBooks at bookboon.com Probability Examples c-4 – Random variables III © 2009 Leif Mejlbro & Ventus Publishing ApS... 978-87-7681-519-6 Download free eBooks at bookboon.com Random variables III Contents Contents Introduction Some theoretical results Maximum and minimum of random variables 20 The transformation formula and... free eBooks at bookboon.com Some theoretical results Random variables III Some theoretical results The abstract (and precise) definition of a random variable X is that X is a real function on Ω,