By Ronald Meester

ISBN-10: 3764387238

ISBN-13: 9783764387235

Compactly written, yet however very readable, attractive to instinct, this advent to chance idea is a superb textbook for a one-semester direction for undergraduates in any path that makes use of probabilistic rules. Technical equipment is simply brought while helpful. The direction is rigorous yet doesn't use degree conception. The textual content is illustrated with many unique and impressive examples and difficulties taken from classical purposes like playing, geometry or graph thought, in addition to from functions in biology, drugs, social sciences, activities, and coding conception. simply first-year calculus is needed.

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**Additional resources for A Natural Introduction to Probability Theory, Second Edition**

**Example text**

34. Suppose we throw a fair coin n times. Let A be the event that we see at most one head, and let B be the event that we see at least one head and at least one tail. Show that A and B are independent when n = 3 and show that they are not independent when n = 3. 35. Suppose that I have two coins in my pocket. One ordinary, fair coin, and one strange coin with heads on either side. I pick a random coin out of my pocket, throw it, and it turns out that head comes up. (a) What is the probability that I have thrown the fair coin?

7. 3, this result leads to the following computation: E(Y ) = x x2 P (X = x) = 1 · 13 + 1 · 13 + 4 · 13 = 2. 8. Show that if X takes only non-negative integer values, we have ∞ P (X > n). E(X) = n=0 For two random variables X and Y on the same sample space, we can deﬁne new random variables X + Y , XY , etcetera via (X + Y )(ω) = X(ω) + Y (ω), and (XY )(ω) = X(ω)Y (ω). 46 Chapter 2. Random Variables and Random Vectors Expectations turn out to behave very nicely under taking sums of random variables.

Hence the gambler is interested in a function of the outcome, rather than in the outcome itself. Such a function is called a random variable and in this chapter we deﬁne and study such random variables. 1 Random Variables Suppose that we ﬂip a coin n times. Each time tails comes up we lose one euro, and when heads comes up we win one euro. The sample space corresponding to this experiment could be {−1, 1}n, the set of sequences of −1s and 1s of length n. How can we express our fortune after n ﬂips?

### A Natural Introduction to Probability Theory, Second Edition by Ronald Meester

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