By R. Meester

"The publication [is] an exceptional new introductory textual content on chance. The classical approach of training chance relies on degree thought. during this publication discrete and non-stop likelihood are studied with mathematical precision, in the realm of Riemann integration and never utilizing notions from degree theory…. a number of issues are mentioned, corresponding to: random walks, susceptible legislation of huge numbers, infinitely many repetitions, robust legislation of huge numbers, branching tactics, susceptible convergence and [the] imperative restrict theorem. the speculation is illustrated with many unique and stunning examples and problems." Zentralblatt Math

"Most textbooks designed for a one-year path in mathematical information hide likelihood within the first few chapters as instruction for the facts to return. This publication in many ways resembles the 1st a part of such textbooks: it is all chance, no information. however it does the chance extra totally than ordinary, spending plenty of time on motivation, rationalization, and rigorous improvement of the mathematics…. The exposition is generally transparent and eloquent…. total, it is a five-star booklet on likelihood that may be used as a textbook or as a supplement." MAA online

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**Extra info for A Natural Introduction to Probability Theory**

**Example text**

The expectation of a random variable will be deﬁned as a sum, possibly with inﬁnitely many terms, and it pays to spend a few lines about such sums. ∞ Let a1 , a2 , . . be real numbers. We only want to deﬁne n=1 an when this sum does not change when we change the order of the an ’s. A classical result − from calculus tells us that this is the case when S + = = n:an >0 an and S + − − n:an <0 an are not both inﬁnite with opposite sign. If S and S are not both ∞ inﬁnite with opposite sign, then we say that the sum n=1 an is well deﬁned.

However, all our sample spaces so far contained at most countably many points, and we shall see later that any sample space which represents inﬁnitely many 38 Chapter 2. Random Variables and Random Vectors coin ﬂips is necessarily uncountable. Is this a problem? In some sense yes, but in another sense no. If we ﬂip a coin, and we are only interested in the time that the ﬁrst head comes up, then it is enough to take {1, 2, . }∪{∞} as our sample space. An outcome k then corresponds to the event that the ﬁrst head appears at the kth ﬂip, and the outcome ∞ corresponds to the event that we never see a head.

11. For any random variable for which E(X) exists and for any a and b, it is the case that E(aX + b) = aE(X) + b. 12. Prove this proposition. Instead of sums, we also need to consider products of random variables. It turns out that for products, independence does play a crucial role. 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.