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b. 1/64 ¢..0137,.0137—d. L817 75. a. 449,699,148 —b. .050, 018
17. b. 90th percentile of Y = 1.8(90th percentile of X) +3277. a.A; —_b. Exponential with 2 = .05
¢. 100 pth percentile of Y = a(100 pth percentile of ¢, Exponential with parameter ni
Pare 83. a. 8257, 8257, .0636 b. .6637—¢. 172.73
Te askensoy tate 87. a..9296 b..2975 98.18,
Bly QsSES2 SEES, bisOHe 89. a. 68.03, 122.09 b..3196 _€. .7257, skewness
23. a. A+ (B— Alp
b. (A + B)2, (B — AP/12, (B — A)/VT2 on a tae aoe *; ae % = Aly
BA" hyn + 1B — AN] . 148. e9.5 125:
25. 314.79 woh
27, 248, 3.6 95. b. P(x + B) Mm + PAP + B +m) V(BYI, Bia + B)
29. 1/(1 = 1/4), 1/4, 16 97. Yes, since the pattern in the plot is quite linear.
31. 1007, 307 By Yes
33. f(a) = dy for —5 <x <5 and = 0 otherwise 101s Yes
38. a. M@ = 180515 — 9,1 < 15: EOD = 7.167, 103. Form a new variable, the logarithms of the rainfall
Voy) = 44.44 values, and then construct a normal plot for the new
b. EO) = 7.167, VON) = 44.44 a Bees of the linearity of this plot, normality
37. M() = SKS — 9, EX) = 6.667, V(X) = 44.44 ;
This distribution is shifted left by .5, so the mean differs 105+ The sora plot lige a.cnoWliness palteni: showitg
by 5 but the variance is the same. Dosivenkenness)
46) iAH BoE GATT BRE 107. The plot deviates from linearity, especially at the low
end, where the smallest three observations are too small
€..9147 £..9599 9104 bh. 0791 AUC REE Ea se ;
i 0668 5.9876 relative to the others. The plot works for any / because /
is a scale parameter.
41. a.134 b.-134 674d. 674 5 sais
e. — 1.555 109%. fy(y) = 2", y > 1
43. a..9772 bd. .9104— 8413 ALO) FEW AEG
2417 f. 6826 113. fy(9) = 16,0 <y < 16
48. a..7977 _ b. .0004 115. fy() = Win +9)
¢. The top 5% are the values above .3987. 5
117. Y=x7/16
47. The second machine
119. fr(y) = 1/2Ve,0<y <1
49. a..2525 b. 39.96 b= eval
121. fr(y) = 1/l4yy, O<¥ <1, fr) = 1/[8VyI,
51. .0510 l<y<d
53. a..8664 = b..0124 2718 125, py(y) = (1 — p)"'p,y = 12.3.0.
--- Trang 834 ---
Chapter5 821
27a4 b.6 eFax) =4/25,0 <x < 25; 9. a. .3/380,000 b..3024—€. 3593
F(x) =0,.x <0; FQ) =1,x>25 12.5, 7.22 d. 10Kx7 + .05,20 << 30 eno
129. b. F(x) = 1 — 16/(x + 4),x > 0; FQ) = 0.x <0 UL. a. p(x,y) = (e428 /x!) (e-" /y!) for x = 0, 1, 2, ..5
©2247 d.4 &. 16.67 y=0,1,2,... b (e*9(1+4+40))
ec. e*"( + 0)"/ml, Poisson with parameter 4 + 0
131. a..6563 b.41.55 3179
ae x >0,y>0 b..3996 5940
133. a. .00025, normal approximation; .000859, binomial a. 3298
b. .0888, normal approximation; .0963, binomial
" 18. a. F(y) =1—2e +e for y > 0, FQ) =0 for
135. a. F(x) =1.5(1 — 1), 1 Sx <3; FQ) =0,x <1; y <0; f(y) =4ie 2 — 3e* for y > 0, fy) = 0
FQ)=1x>3 b.9%4 ©. 1.6479 fory <0
d..5333 e. .2662 b. 2/62)
137. a. 1.075, 1.075 b..0614,.3331 «2.476 i.a25 bln en
139. b. 95,693, 1/3 d. fx(x) = 2VR7— 22 /(R?) for, -R <x <R,
2 2 f(y) = 2,\/R? — y2/(R?) for -R < y < Ry no
141. b. FQ) = Se, x < 0; FQ) = 1 — Sex > 0
€. 5, .6648, .2555, .6703 19. .15
143. ak=(2- 15"! db. F(X) =0,x <5; 21. L?
=1— (Six) a =
Fa) =1- GIN x >5 eS DIE) 95 yy
145. b. 4602, 3636 €..5950 140.178 2528
2
147. a. Weibull by 5422 27. a, —10588 —_b. ~.0128
149. ai bea 1p
@ F(t) 1—e #e *10P),0 <x < fia) =0, 3% afl) = 2x, 0 <x < 1, fs) = 0 elsewhere
EOF =e D.fnxGlay =U O<y<x<l 6.6
: 2(s-"/09)) d. no, the domain is not a rectangle
f(x) = a(1 — x/B)e Y.05x SB e. E(YIX = x) = x/2, a linear function of x
fix) = 0,x < 0, fx) = 0.x > B £ VX =x) = x12
This gives total probability less than 1, so some ;
probability is located at infinity (for items that last 39. a. f(x) = 2e°**,0 <x < 00, f(a) = 0.x < 0
forever). b. fxs) = "0 <x < yy < 00
PY > 2x = 1) = Ve
151. jue ~ v/20, x © v/800 d. no, the domain is not rectangular
7 e. (VIX = x) =x + 1,a linear function of x
155. Fgh) = 818 £VYX=x)=1
41. a. EM X =x) = 1/2, a linear function of x; V(YI
Chapter 5 X=y=r/2
b. fit, y) = 1k, 0<y <x
1. a. .20 b. 42 . The probability of at least one ¢. fy) = —Iny),0<y <1
hose being in use at each pump is .70. a. EQ) = 1/4, VX) = 7/144
dx o 1 2 y» oOo 1 2 e. EY) = 1/4, VY) = 7/144
px) | 163450 pv) 24383843. a. py (OIL) = 4/17, pyx(IIL) = 10/17, pyx(2I) = 3/17
PX <1) =.50 b. prx(O12) = -12, prx( 112) = 28, prx(2I2) = .60
e. dependent, .30 = P(X = 2 and Y = 2) 4 P(X = 2) ec: 40
P(Y = 2) = (.50)(.38) d. pyy(012) = 1/19, pyy(112) = 3/19, pyr(212) = 15/19
BadS b40 © .22=P(A)=PUX,-XA>2) 4 aAMX=y=x2 bVNX=y=x°/12
d...17, 46 efx) =y?-10<y<l
eon 0 1 2 3 447. a. pls!) = p22) = pB3) = 19, p21) = pBA)
pices) 19 30 25 4 2 = p32) = 2/9
E(X;) = 1.7 b. px(1) = 1/9, px(2) = 3/9, px(3) = 5/9
¢. pyw(IIL) = 1, pyx(1I2) = 2/3, pnx(2I2) = 1/3,
Brugge 0 1 2 3 Prx(13) = 4, prx(213) = -4, prx(3[3) = .2
2+ SS d. AX = 1) = 1, BX = 2) = 473,
Pal%2) 19 30 28 23 E(VX = 3) = 18, no
8 0 =p(4 , 0) A pi(4) - px(0) = (.12)(.19) so the two e. VOX = 1) = 0, VINX = 2) = 2/9,
variables are not independent. VINX = 3) = 56