By R. Meester
"The e-book [is] a good new introductory textual content on chance. The classical manner of educating likelihood is predicated on degree thought. during this e-book discrete and non-stop likelihood are studied with mathematical precision, in the realm of Riemann integration and never utilizing notions from degree theory…. quite a few themes are mentioned, similar to: random walks, susceptible legislation of enormous numbers, infinitely many repetitions, powerful legislation of huge numbers, branching techniques, vulnerable convergence and [the] vital restrict theorem. the speculation is illustrated with many unique and outstanding examples and problems." Zentralblatt Math
"Most textbooks designed for a one-year direction in mathematical statistics disguise chance within the first few chapters as education for the records to come back. This ebook in many ways resembles the 1st a part of such textbooks: it is all likelihood, no records. however it does the chance extra absolutely than traditional, spending plenty of time on motivation, clarification, and rigorous improvement of the mathematics…. The exposition is generally transparent and eloquent…. total, it is a five-star publication on chance which may be used as a textbook or as a supplement." MAA online
Read or Download A Natural Introduction to Probability Theory PDF
Best probability books
This ebook is set stochastic-process limits - limits within which a series of stochastic procedures converges to a different stochastic method. those are necessary and fascinating simply because they generate uncomplicated approximations for classy stochastic methods and in addition support clarify the statistical regularity linked to a macroscopic view of uncertainty.
A vintage textual content, this two-volume paintings offers the 1st whole improvement of chance idea from a subjectivist point of view. Proceeds from an in depth dialogue of the philosophical and mathematical points of the principles of chance to a close mathematical remedy of chance and information.
Very good finished monograph on multivariate t distributions, with quite a few references. this can be the one publication focusing solely in this subject that i am acutely aware of.
similar in caliber and intensity to the "discrete/continuous univariate/multivariate distributions" sequence via Samuel Kotz, N. Balakrishnan, and Norman L. Johnson.
- Probability and Partial Differential Equations in Modern Applied Mathematics
- Exploring Probability in School: Challenges for Teaching and Learning (Mathematics Education Library)
- Credit risk: modeling, valuation, and hedging
- Large deviations
- Journées de Statistique des Processus Stochastiques: Proceedings, Grenoble, Juin 1977
Additional resources for A Natural Introduction to Probability Theory
The random variable X represents the number of heads when we ﬂip a coin n times, where each ﬂip gives heads with probability p. In particular, we can write such a random variable X as a sum n Yi , X= i=1 where Yi = 1 if the ith ﬂip yields a head, and Yi = 0 otherwise. It is clear that E(Yi ) = p, and according to the sum formula for expectations, we ﬁnd that E(X) = np. 33. Use the sum formula for variances to show that var(X) = np(1 − p). Here we see the convenience of the sum formulas. Computing the expectation of X directly from its probability mass function is possible but tedious work.
13. Find two random variables X and Y so that E(XY ) = E(X)E(Y ). 14. If the random variables X and Y are independent and E(X) and E(Y ) are ﬁnite, then E(XY ) is well deﬁned and satisﬁes E(XY ) = E(X)E(Y ). Proof. We write lP (XY = l) = l l l l ) k lP (X = k, Y = l ) k k = k P (X = k, Y = l = lP (X = k)P (Y = k = l kP (X = k) k=0 = l=0 l ) k l l P (Y = ) k k E(X)E(Y ). Hence the sum in the ﬁrst line is well deﬁned, and is therefore equal to E(XY ). 15. Let X and Y be independent random variables with the same distribution, taking values 0 and 1 with equal probability.
The number of heads is a random variable which we denote by X. 1. Random Variables 37 for k = 0, . . , n, and pX (k) = 0 for all other values of k. Hence its distribution function is given by n FX (x) = 2−n , k 0≤k≤x for 0 ≤ x ≤ n; FX (x) = 0 for x < 0; FX (x) = 1 for x > n. 7 (Binomial distribution). A random variable X is said to have a binomial distribution with parameters n ∈ N and p ∈ [0, 1] if P (X = k) = n k p (1 − p)n−k , k for k = 0, 1, . . , n. We have seen examples of such random variables in Chapter 1 when we discussed coin ﬂips.
A Natural Introduction to Probability Theory by R. Meester