An Intermediate Course in Probability
The purpose of this book is to provide the reader with a solid background and understanding of the basic results and methods in probability theory before entering into more advanced courses (in probability and/or statistics). The presentation is fairly thorough and detailed with many solved examples. Several examples are solved with di erent methods in order to illustrate their di erent levels of sophistication, their pros, and their cons. The motivation for this style of exposition is that experience has proved that the hard part in courses of this kind usually is the application of the results and methods; to know how, when, and where to apply what; and then, technically, to solve a given problem once one knows how to proceed. Exercises are spread out along the way, and every chapter ends with a large selection of problems. Chapters 1 through 6 focus on some central areas of what might be called pure probability theory: multivariate random variables, conditioning, tra- forms, order variables, the multivariate normal distribution, and convergence.
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branching process central limit theorem characteristic function Compute conclusion conditional distribution conditional expectations converges in distribution corresponding deﬁned Deﬁnition density function Determine the distribution diﬀerent Diﬀerentiation distribution function emitted equals Example 3.1 Exercise exist exponential distribution fact ﬁnd Find the distribution ﬁnite ﬁrst ﬁxed formula fX(x fY(y given gX(t i.i.d. random variables iﬀ independent random variables inﬁnitely joint distribution large numbers law of large lemma Let X1 limit distribution linear martingale mean moment generating function nonnegative normal distribution obtain occurrence order statistic otherwise P(Xn parameter particles Poisson distribution Poisson process probability theory problem process with intensity proof of Theorem prove r-mean random vari random vector Remark result sample sequence of random set Sn Show that Xn stochastic process summands symmetric Theorem 2.1 total number transforms U(0,1)-distributed random uniformly variance VarX Xn converges yields zero