# Probability and Random Processes One Thousand Exercises in Probability

The copyright of all linked material is owned by Geoffrey R. Grimmett and David R. Stirzaker, 2 May 2002.

Revised pages, second printings, Summer 2002

(PRP) 12, 113, 243, 290, 545, 553, 577-580.

(OTEP) 2-3, 129, 138-139, 271, 341, 414.

We are grateful to readers for their suggestions for improvements.

• page 203 of PRP: line -4: 5.7.11, not 5.8.11.
• Problem 3.11.36: \mu = \bar x. The superscript is incorrect in the first summation. The problem is best solved by permuting the X_i at random first.
• Problem 5.12.38: The solution in OTEP to the last part is (at best) misguided. Take 0 < p < 1 and 0 < \beta < \beta_c. The mean number of paths, length n, from the root satisfies 2^n P(S_n \ge \beta n) \to \infty, so there exists k with 2^k P(S_k \ge \beta k) > 1. Define k accordingly. Now construct a branching process as follows. From the original tree T construct a tree T_k whose vertices are those of T at distance rk from the root, with r=0,1,2,\dots. For u,v \in T_k, connect u and v by an edge iff the path from the root to v passes through u, and the distance from u to v in T is k. For such u, v, we declare the edge from u to v to be gray' if the unique T-path \pi from u to v contains at least \beta k black vertices. The mean number of gray edges leaving u downwards is 2^k P(S_k \ge \beta k) > 1. Thinking of the green edges as denoting the existence of children in a branching process, the mean family-size is > 1, so this process survives with strictly positive probability. That is, with probability > 0: for all r, there exists a T-path from the root, length rk, with at least \beta rk black vertices. The final argument given in OTEP requires a fix also.
• page 409 of PRP: on the last line but one, X_0 should be X_1.
• Problem 9.7.15: In the solution of OTEP to Problem 9.7.15, on line 4 of p 366 (et seq) there should be no minus sign in the lest exponent.
• page 477: The first occurrence of mu in Example 12.2.8 should be mu^2.
• page 482 of PRP: Delete the expectation from equation (4); add to LHS of final display. Replace lemma' on last line by theorem'.
• Problem 12.9.15: Replace largest stake' by `maximum deficit'. In the solution, the random walk has mean step size -1/2.
• pages 523-524 of PRP: the proof implies the UNIFORM continuity of X on the dyadic rationals and this is needed in extending the definition to the reals.

# Note: all the above corrections (and more) to PRP have been implemented in the reprint of September 2008.

14 December 2010: Thank you, readers, for writing with errata and suggestions, and special thanks to Peter Ralph and friends at Berkeley, and to Jerry Zhao. Minor changes have been made in the 2011 reprints of PRP and OTEP, on the following pages:

• PRP: 44, 87, 161, 175, 318
• OTEP: 14, 49, 51, 83, 86, 105, 156, 238, 294, 295, 325, 337, 364, 435
Detection of most of these will require a microscope (though Exercises 5.4.2 and 7.2.5(b) were simply incorrect).