By R. Meester

ISBN-10: 3764387238

ISBN-13: 9783764387235

ISBN-10: 3764387246

ISBN-13: 9783764387242

"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

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Additional resources for A Natural Introduction to Probability Theory

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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.

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A Natural Introduction to Probability Theory by R. Meester

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