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P(GIRs < Ry < R2) = 1/15, so classify as basalt if UUUAA) = .00324 yoo
Ry < Ry < Ro. 4. PY =y) = — ICIP. y = 23,45...
e175 d.p > 14/17 19. ce. F(x) = 0, x <1, FQ) = logio(X] +1), Sa <9,
101. a, 1/24 b. 3/8 FQ) =1,x> 9
d. 602, 301
103. s=1
2. F(X) = 0, x <0; 10, 05x <1; 25,1 Sx <2; 45,
107. a, P(Bolsurvive) = bo/[| — (by + bred] 2Sx<3; 70, 35e<4 90, 45x<5; 6,
P(By\survive) = b,(1 — ed)/{1 — (by + bed] 5 <x <6; 1.00,6 <x
P(Balsurvive) = bx(1 — ed)/{1 — (by + ba)ed]
b. 712, 058, 231 23. a. p(l) =.30, p3)=.10, p(4) =.05, p(6) = 15,
pl2) = 40
b. .30, .60
25. a. p(x) = (1/3)(2/3)""} x = 1, 2,3...
b. py) = (1/3)(2/3"", y = 2, 3,4,
. pO) = 1/6, pl) = (25/5449) 12 = 1,2,3,4,...
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Chapter3 819
29. a. 60 b. $110 85. a. h(x; 10, 10,20) b. 0325. ACs m,n, 2n),
E(X) = n/2, VX) = 1° /f42n — 1)]
31. a. 16.38, 272.298, 3.9936 b.401 ¢. 2496 d. 13.66
7 87. a. nbQs 2, 5) = (+ 1.5%, x = 0, 1, 2,3...
33. Yes, because E(1/a*) is finite. b. 3/16 ©. 11/16 2,4
35. $700 89. nb(x; 6, 5), EX) = 6 = 3(2)
37. E{h(X)] = 408 > 1/3.5 = .286, so you expect to win 93 9932, .065 068d 491s 251
more if you gamble.
95. a..011 b.441 6. .554,.459 944
39. V(-X) = VX)
97. a..491— b..133
AL. a, 325 b.7.5
¢. VO) = EIX(X-1)] + EQ) — (E00? 99. a. 122, .808,.283 b, 12,3.464 e530, 011
43. a. 1/4, 1/9, 1/16, 1/25, 1/100 101. a..099b..135 2
b. w= 2.64, o = 1.54, P(X — pl > 20) = 04 < 25,
P(X — pl > 30) -0< 1/9 103.4 b..215—e. 1.15 years
The actual probability can be far below the Chebyshev 495, 9,221, 6.800.000 _¢. pla: 1608.5)
bound, so the bound is conservative.
¢. 1/9, equal to the Chebyshev bound LIL. b. 3.114, 405, .636
d. P(—1) = 02, P(0) = .96, P() = .02
113. a. (x; 15,.75) b. 6865 @..313 d. 45/4, 45/16
45. Mx(0) = Se'M(1-.Se"), BX) = 2, VX) = 2 e309
47. py(y) = 75.251, y = 1,2,3,.- 115. .9914
49, E(X) = 5, V(X) = 4 117. a. px; 2.5) b..067—c. .109
51. My(t) =e" 7, EX) =0, VX) =1 119. 1.813, 3.05
53. E(X) = 0, V(X) = 2 121. p(2) = p®.p@) = U1 — pp? .p(4) = - Pp,
= [1 = p(2) — +++ — pix — 31 — pp, x =5,
59. a..850 b..200 200 d..701 pel “E i PE SE — PM
e851 (£000 gs 570 Alternatively, p(x) = (1 — p)pix — 1) +
61. a..354 0b. 114 e919 PCL — p) p(x — 2),x = 5, 6, 7, ... 3 99950841
@ ado? be 8 123. a. 0029 b..0767,.9702,
65. 1478 125. a..135 b..00144 ee. & bes2yF
67. .4068, assuming independence 127. 3.590
69. a. 0173 b. .8106, 4246. 0056, 9022, 5858 WesasNo: bis02TS
71. For p =.9 the probability is higher for B (.9963 versus 131. b..6p03 4) + Apo, eA + w/2
.99 for A) (4+ wi2+ — wl
For p = 5 the probability is higher for A (.75 versus 433, 5
.6875 for B)
Fir chuolinlationtieaes obenenedied 137. X ~ b(x; 25, p), E(W(X)) = 500p + 750,
. The tabulation for p > .5 is not neede
f ux) = 100\/p(1 — p)
75. a. 20, 16 (binomial, n = 100, p = 2) b. 70, 21 Independence and constant probability might not be
valid because of the effect that customers can have on
77. When p =.5, the true probability for k = 2 is .0414, each other. Also, store employees might affect customer
compared to the bound of .25. decisions,
When p = 5, the true probability for k = 3 is .0026,
compared to the bound of .1111. 139.
When p = .75, the true probability for k = 2 is .0652, x 0 1 2 3 4
compared to the bound of .25. p(x) 07776 10368 19008 20736 17280,
When p = .75, the true probability for k = 3 is .0039,
compared to the bound of .1111. x 5 iJ id 8
PQ) 13824 06912 03072 01024
79. Myx) =[p+(1— pel", Ew — X) =n ~ p),
Vin — X) = np(l = p)
Intuitively, the means of X and n — X should add to n and
their variances should be the same,
81. a..114 —b..879 e121 Use the binomial
distribution with n = 15 and p = .1
83. a. h(x; 15, 10,20) b..0325—&. .6966
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820 Chapter 4
Chapter 4 55. a..794 b.S88 7.94 265
57. No, because of symmetry.
ads bs «7/16
59. a, approximate, .0391; binomial, 0437
3. b.S e 11/16 d. 6328 b. approximate, .99993; binomial, .99976
5.a.3/8 d.1/8 2969 d..S781 61. a..7287 _b, .8643, .8159
7. a. f(x) = qyfor 25 < x < 35 and = 0 otherwise 63. a. approximate, .9933; binomial, .9905
b2 G4 2 b. approximate, .9874; binomial, .9837
® as8 ae, OO O70 ¢. approximate, .8051; binomial, .8066
67. a. 15866 _b. .0013499 _¢, .999936658
Ll. a. 1/4 b. 3/16 ©. 15/16 5
iB e hocah beoeees, wt ten Actual SB ag ID 999936658
otherwise -
13 4.3. OEE <a ates 4 690.120 1329 e371 d.735 0
c. 1/8, .088 Ta.54 bTIS All
15. a. F(x) = 0 forx < 0, Fx) = 7/8 for 0 <x <2, Bal bl ©9982 4.129
FQ) = 1 forx > 2