\magnification=\magstep1 \def\mn{\medskip\noindent} \def\bn{\bigskip\noindent} \parindent=0pt \parskip=3pt \mn {\bf ESSENTIALS OF PROBABILITY Typos, October 20, 1996} \mn First for the book and then for the Instuctor's Solution manual. For the students solution manual take the appropriate subset of the latter. \mn {\bf Chapter 1} p.12 Exercise 2.11. $2+4+4$ appears twice and $2+2+6$ is omitted. p.22 Exercise 3.35. $=n(n-1)2^{n-2}$ p.22 Exercise 3.37. This only holds if $n>1$. p.22 Example 3.6. into $m$ distinguishable groups \mn p.26 Example 4.4. Let $X$ be the number of defectives in the population and $G$ the event that there is no defective. In symbols we compute $P(G|X = 50)$ which gives an upper bound on $P(G|X \ge 50)$. However then the book gets confused and makes claims about $P( X \ge 50|G)$. This will have be rewritten. \mn p.44 Exercise 6.17. $p_0(r,n) = \sum_{j=0}^r (-1)^r$. In the defintion of $p_k(r,n)$, use $k/r$ not $k/n$ p.47 Eexercise 7.3. Place eight rooks on the chessboard. p.48 Exercise 7.17. (c) all 5 families \mn {\bf Chapter 2} p.56 Exercise 1.18. Assume the students are independent p.57 Exercise 1.25. the probability of having at least two boys p.57 Exercise 1.27. not draw a spade? p.58 Example 2.3. $P(A)=13/52$ p.69 Exercise 3.11. can narrow the choices down to two in 3 cases p.77 Exercise 4.3. the fractions are $p_O^2$, $2p_Ap_O+p_A^2$, $2p_Bp_O+p_B^2$ and $2p_Ap_B$. p.84 Exercise 5.11. $\alpha=1/\sum_{i=0}^{N-1} P(X>i)$ if $P(X>N)>0$. p.91 Exercise 6.2. Suppose $A^c=B \cup C$ \mn {\bf Chapter 3} p.98 Exercise 1.4 asks about the exponential distribution before it is defined on page 101. p.98 Exercise 1.7. the Poisson approximation of $P(X=1)$ p.99 Exercise 1.17. $n=\sqrt{2\lambda m}$ p.100 Delete stray ``in'' in first sentence below graph p.108 Exercise 2.16. $(2\pi)^{-1/2}(x^{-1}-x^{-3})e^{-x^2/2} \le 1-\Phi(x) \le (2\pi)^{-1/2} x^{-1} e^{-x^2/2}$ p.108 Exercise 2.17. Show that $P(M \ge m+n|M \ge m) = P(M \ge n)$. The geometric distribution we have defined has $P(M > m+n|M > m) = P(M > n)$. p.113 Assume $\lambda > 0$ in Exercise 3.2; $n>0$ in 3.3; $\alpha>0$ in 3.5 p.114 Exercise 3.7. Suppose $X$ has the standard normal distribution p.117 line 7: $(X,\Theta)$ has joint density p.122 Example 5.1. $Y$ values are 100, 125, 150, 175 not 0, 1, 2, 3. p.129 Exercise 5.9. $f_{(Y,Z)}(y,z)=1/2y$ p.133 Exercise 6.4. $\Theta=\tan^{-1}(X_2/X_1)$ p.134 Exercise 6.6. $X_2=\sqrt{-2\log U_1}\sin(2\pi U_2)$ p.134 Exercise 6.8 the $k$th smallest of p.137 Example 7.4. and $X_2 = \hbox{normal}(\nu,b)$ are independent then p.139 Exercise 7.7. $T=X_1+ \cdots + X_n$ p.150 Exercise 9.16. $Z=X/\sqrt{Y/m}$ \mn {\bf Chapter 4} p.159 Exercise 1.17. $cx^{-2}(\ln x)^{-a}$ for $x\ge e$ p.160 Exercise 1.22. $EX \approx Nn/(n+1)$ when $N$ is large and $n/N$ is small p.167 Exercise 2.14. $\phi_X(t) = \sum_{k=0}^\infty {t^k \over k!} \phi_X^{(k)}(0)$ p.167 Exercise 2.26. $Ef(N)=$ p.175 In (3.6) and (3.7) need to assume that the expected values exist p.180 Exercise 3.22. sum over $i_1< i_2 < \ldots i_k$ p.192 Exercise 4.9. $v(X) = \sqrt{EX^2}/EX$ p.193 Exercise 4.22. $R=Z_1+ \cdots + Z_{m+n}$ p.202 Exercise 5.11. $X_1, X_2, X_3$ are independent p.214 Exercise 6.19. line 4: $P(\xi_{n,m}=k)=p_k$; line 11: Use (6.11) and (a) p.214 Exercise 6.20. $N_n$ not $Z_n$ (twice) p.214 Exercise 6.21. $a=1-\{bc/(1-c)\}$ where $b=0.5$ and $c=0.6$. p.219 Exercise 7.6. for $r\ge 0$ p.220 Exercise 7.19. the number of boxes you will p.220 Exercise 7.25. $-x 0.05 \}$ \bn {\bf Answers to Selected Exercises}, i.e., pages 257--262 \mn {\bf Chapter 1} 4.10. $C_{16,5} = 4386$ (a) 1792/4368, (b) 2240/4368, (c) 336/4368 4.14. $426/924 = 0.5$. 7.20 (a). 0.7969. 7.22. 20/32 \mn {\bf Chapter 2} 3.12. 0.6. 4.14 ,(b). 3/7. 4.18. 8/9 5.2. (a) 0.2244, (b) 0.0016. 5.6. $5.9 \times 10^{-6}$ \mn {\bf Chapter 3} 2.1. (a) 0.8399 2.16. $(35/216)(2\pi)^{-1/2} e^{-18} \le 1 - \Phi(6) \le (1/6)(2\pi)^{-1/2} e^{-18} = 1.01 \times 10^{-9}$ \mn {\bf Chapter 4} 4.30. $-(nKL/N^2)\{1 - (n-1)/(N-1)\}$. 5.16. $1\le \hbox{var}(X+Y) \le 9$ 6.2. 0.9732. 7.8. $(n-1)/(n+1)$. 7.18. variance 11.55 \mn {\bf Chapter 5} 2.8. 0.0618. 3.12. $[2.7,3.3]$. 5.6. $[8.76,13.24]$ \bn {\bf Typos in INSTRUCTORS SOLUTION MANUAL} \mn {\bf Chapter 1} p.1, 1.5, (c): $A \cap (B \cup C)$ p.6, $7 \cdot 2 \cdot 5!$ p.7, 3.31, (a): divide answer by $4!$ to account for the fact that it does not matter in which order the couples are chosen. p.9, 4.3, (b): $C_{13,2} \cdot C_{4,2} \cdot C_{4,2} \cdot 44$, pick the values for the pairs, assign suits to the values then pick the last reamining card p.11, 4.17: $(4 \cdot C_{13,2} \cdot 13^3)/C_{52,5})$, since some suit will appear twice and the other three once each. p.13, 6.1: delete $\{$ and $\}$ for the one $A_{\{i,j\}}$ p.15, 6.11: $-0.0462$ p.18, 7.5: $11!$ instead of $(11-k)!$ in denominator; $P(\cup_{i=1}^6 A_i)$ p.18, 7.6: $P_{26,4} - P_{21,4}$ p.19, 7.9: if and only if no outcome in exactly one set. (The solution forgets about the possibility of being in 0 sets.) p.21, 7.23, (d): replace $C_{10,3}$ by $40 = C_{10,1} \cdot 4$. First term is exactly two pair, the second exactly three pair. \mn {\bf Chapter 2} p.22, 1.9: $7/36 = (1/3) \cdot (21/36)$ p.26, 2.10: 0.4275 p.30, 3.17: $P(C) = 1/4$ not 1/3 so $P(W) = (1/4)/\{ 1- (3/4)(2/3) \} = 1/2$ p.31, 4.2: If we are picking children at random then $P(B_i) = i/6$ and the answer changes to $$ P(B_1|A) = { 1/12 \over 1/12 + 1/6 + 3/16 } = { 8 \over 8 + 16 + 18 } = {4\over 21} $$ p.40, 6.15: $P(A_{14})$ \mn {\bf Chapter 3} p.48, 3.8: right hand side should be $1/\pi(1+x^2)$ p.51, 4.5: For $x,y\in\{1,2,3,4\}$, $P(X=x, Y=2x) = 1/16$ while $P(X=x,Y=y) = 2/16$ if $x \le y < 2x$ p.53 line 6: $r^2\sin\phi$ \bye