Probability and Random Processes, 4e

One Thousand Exercises in Probability, 3e

by Geoffrey Grimmett and David Stirzaker

Published in 2020 by Oxford University Press. [Note added November 2024: these two volumes should now be available after recent revision to fix minor slips and to add a few new exercises.]

What's new in this pair of volumes?

• Reorganization and expansion of certain material.
• Addition of around 300 new exercises and problems, making a total of over 1300.
• Inclusion of new sections on material including:
  • coupling from the past,
  • Lévy processes,
  • self-similarity, stability, and time changes,
  • an expanded treatment of continuous-time Markov chains, via the holding-time/jump-chain construction.

The solutions to all exercises and problems have been written up in the third edition of One Thousand Exercises in Probability.

The copyright of all linked material rests with the authors.

Preface and Contents of PRP4e.

Typos/errors:

• OTEP, solution to (1.8.30). n should be m in first display
• OTEP, solution to (3.11.50d). Let X and Y be independent. Then H(X|Y) = H(X).  Since the entropy of X is unchanged by any constant translation y to X + y, we have H(X + Y|Y) = H(X|Y).
  Now let Z_n be bin(n, p), independent of B which is Bernoulli ber(p).  By the result of Exercise 3.6.5,  
  H(Z_n+1) = H(Z_n + B) >= H(Z_n + B|B) = H(Z_n|B) = H(Z_n).
  Note to reader: You could spend a little while trying to prove this directly from the explicit expression H(Z_n) = nH(B)-E(log(n \choose Z_n )). And, naturally, the entropy nH(B) of n Bernoulli trials also increases with n; the difference being the amount of information in the positions of the Z_n successes. [Thanks to Joseph Boutros for pointing out the error.]
• PRP+OTEP, Exercise 4.2.5. The answer is wrong, and should be the sum of 1/k from k=1 to k=n. The solution is (inevitably) wrong also. Using the linearity of expectation, the answer is n times the integral over [0,1]^2 of the function (1-xy)^{n-1}. Integrate over y, and change variable a=1-x, to obtain the integral over [0,1] of (1-a^n)/(1-a). Expand the integrand as a finite geometric series, and it is done. [Thanks to Julius Plenz for pointing this out.]
• Exercise 4.4.7(a). Replace beta-1 by beta in the question.
• Exercise 4.4.10. The given answer is incorrect and should be (1/lambda) + (1/lambda^2) (e^{-lambda} - 1). [Thanks again, Julius.]
• PRP, Example (5.2.5). Delete the word "colour" on line 16 of this page.

PRP. Many thanks to Peter Stahlecker for pointing out the following.

• p 133. "If (8) holds..." should refer to (7)
• p 136. The reference should be to example (4.8.4) not to (4.4.4)
• p 229, l 18: M should be M^c
• p 290, last display. Extraneous comma in subscript
• p 294, line 6 fb. Should be t >0
• p 306, final display. Add equation number (1)
• p 324, l 2. Subscript n should read k
• p 337, footnote. Exercise number is (3.11.18b)
• p 356, l 26. First X should be X_n
• p 358. (5.9.4) is a definition not a theorem
• p 375. Corollary (6.4.22) should be (6.4.25)
• p 386, l 2, and p 398, l 8. Fatou's Lemma is (5.6.14) not (5.6.13)
• p 394, l 5. Should be Exercise (7.9.4c) not (7.9.4iii)
• p 399, l 7. Should be Exercise (7.9.4f)
• p 430, l 3 fb. S should be T
• p 443, l 10, Example (6.10.12) should be (6.10.6)
• p 444, l 12. The correct reference is Problem (8.10.2)
• p 471. Should be Definition (6.8.16)
• p 488, l 5 fb. Theorem (6.4.21) should be (6.4.24)
• p 504. Equation (14) should be (15)
• p 505, l 7. (6.11.12) should be (6.11.18)
• p 508, l 8 fb. Theorem (6.4.17) should be (6.4.20)
• p 510, l 9. Figure 5.1 should be 5.2
• p 528, l 2. if and only if
• p 604, l 15. Missing 1/2 before W_t²
• p 610, l 6 fb. Missing footnote, which should read: Here $f_t(t,X)$ and $f_x(t,X)$ denote the derivatives of $f$ with respect to its first and second arguments respectively, and evaluated at $(t,X_t)$.
• p 615, top paragraph. This would benefit from rewriting.