By R. Meester

"The e-book [is] an exceptional new introductory textual content on likelihood. The classical approach of training chance relies on degree thought. during this booklet 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 subject matters are mentioned, resembling: random walks, susceptible legislation of enormous numbers, infinitely many repetitions, powerful legislation of enormous numbers, branching methods, vulnerable convergence and [the] critical restrict theorem. the speculation is illustrated with many unique and astounding examples and problems." Zentralblatt Math

"Most textbooks designed for a one-year direction in mathematical information conceal likelihood within the first few chapters as coaching for the information to return. This ebook in many ways resembles the 1st a part of such textbooks: it is all likelihood, 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, this can be a five-star ebook on likelihood that may be used as a textbook or as a supplement." MAA online

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

**Example text**

We shall often write {X = x} for {ω : X(ω) = x} and P (X = x) for P ({ω : X(ω) = x}). 2. The probability mass function of a random variable X is the function pX : R → [0, 1] given by pX (x) = P (X = x). 3. The distribution function of a random variable X is the function FX : R → [0, 1] given by FX (x) = P (X ≤ x). 4. Suppose that the random variable X takes the value 1 with probability 1, that is, P (X = 1) = 1. The distribution function is then given by FX (x) = 1, for x ≥ 1, and FX (x) = 0 for x < 1.

This example shows that a family can be pairwise independent without being independent. 7. This example is classical and bound to surprise you. Suppose we have r people in a room. We assume that their birthdays are equally likely to be any day of the year (which we assume to have 365 days, ignoring leap years). Furthermore, we assume that the events that the birthday of a particular person is on a particular day, are independent. What is the probability that no two persons in the room have the same birthday?

Are A and B independent? 29. We take two cards from a regular deck of 52 cards, withour replacement. A is the event that the ﬁrst card is a spade, B is the event that the second card is a spade. Are A and B independent? 30. We choose a month of the year so that each month has the same probability. Let A be the event that we choose an ‘even’ months (that is, februari, april, . ) and let B be the event that the outcome is in the ﬁrst half of the year. Are A and B independent? If C is the event that the outcome is a summer month (that is, june, july, august), are A and C independent?