Problems in Probability

The following sections are included:

• Preface

• Contents

The following sections are included:

• Notes

• Problems

The following sections are included:

• Notes

• Problems

The following sections are included:

• Notes

• Problems

The following sections are included:

• Notes

• Problems

The following sections are included:

• Notes

• Problems

The following sections are included:

• Notes

• Problems

(1) It is not good teaching practice to start off with an advanced problem. However, this problem is particularly intriguing because, although it deals with the most elementary notions of set theory, it throws you immediately into a fascinating topic in mathematics namely the study of Dedekind numbers) in which there is much activity. It leads you also to the wonderful world of sequences…

(1) A table of this type is often called a "contingency table" as it depicts all possible contingencies of the survey outcomes…

(1) Let $\zeta \left(s\right)={\displaystyle \sum _{t=1}^{\infty }}{t}^{-s}$ for s > 1. Note that for s ≤ 1, this series is divergent.

(a) Suppose that the distribution of X is given by $\begin{array}{cc}& P(X=t)={\left(\zeta \right(2\left)t^2\right)}^{-1}(t=\mathrm{1,2}\mathrm{,3},\dots ).\hfill \end{array}$ This is a bona fide distribution because P(X = t) > 0 for t = 1,2,3, … and ${\displaystyle \sum _{t=1}^{\infty }}P(X=t)=1$. However ${\displaystyle \underset{t=1}{{\displaystyle \sum ^{\infty }}}}tP(X=t)=\zeta {\left(2\right)}^{-1}{\displaystyle \underset{t=1}{{\displaystyle \sum ^{\infty }}}}{t}^{-1}$ and hence μ_X are not defined…

(1) For P > 1, ${\displaystyle \underset{1}{\overset{\infty }{\int }}}{t}^{-p}dt=1/(p-1);$ for p ≤ 1, this integral does not exist…

(1) This is a long question which takes you through the steps of proving this important result. Although the result is important in probability theory, the proof is often omitted from probability texts. I hope that you enjoy the journey which is typical of the reasoning used in proving similar results in classical analysis…

You will notice that the style of arguments used here are very different from the styles of arguments which you encounter in calculus and algebra. In those subjects there is a mechanical approach to many problems (… differentiate both sides, solve for dy/dx, put dy/dx = 0 and solve …).

On the other hand, in this chapter we set out our arguments in a more descriptive fashion; these sorts of arguments bring your analytical skills up to a higher plane. Also, you see that mathematics deals with ideas, concepts and arguments — not merely calculation…

The following sections are included:

• Bibliography

• Index