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