1.1-2 (a) S = {bbb, gbb, bgb, bbg, ggb, gbg, bgg, ggg} (b) S = {F, M}; (c) S = {000, 001, 002, ..., 999}
1.2-2 0.09
1.2-8 (a) 0.6; (b) 0.1; (c) 0.7
1.3-2 Permutation: 4P3 = 24
1.3-6 (a) 80; (b) 256;
1.3-10 20
1.3-12 36,864
HW 2:
1.4-2 (a) 0.71497; (b) 0.61927; (c) 0.78858
1.4-4 (a) 0.058824; (b) 0.063725; (c) 0.019231
1.4-8 0.16975
1.5-2 (a) 0.72; (b) 0
1.6-2 (a) 0.79; (b) 0.43038
1.6-6 P(Stand | Die) = 0.65934; P(Prefered | Die) = 0.26374; P(Ultra-prefered | Die) = 0.076923;
1.6-8 P(B1 | Repair) = 0.63492; P(B2 | Repair) = 0.23810; P(B3 | Repair) = 0.095238; P(B4 | Repair) = 0.031746
2.1-2 (a) f(1) = 0.6, f(5) = 0.3, f(10) = 0.1;
2.1-8 (a) f(0) = f(1) = ... = f(7) = 1/8;
2.1-10 (a) P(X = 1) = (3C1 * 47C9)/(50C10), (b) (3C0 * 47C10)/(50C10) + (3C1 * 47C9)/(50C10);
2.2-2 E[X] = 0, E[3X^2 - 2X + 4] = 20/3;
2.2-4 - $0.50
2.2-6 c = 6/147, E[X] = 12/49;
2.3-2 (a) E[X] = 3/4, E[X(X-1)] = 3/8, Var(X) = 9/16; (b) E[X] = 2, E[X(X-1)] = 3, Var(X) = 1
2.4-2 the pmf of X: f(1) = 7/18, f(-1) = 11/18; the mean of X: E[X] = -2/9, Var(X) = 77/81
2.4-6 (a) Binomial distribution: b(7, 0.15) (b) (i) P(X >= 2) = 0.28342, (ii) P(X=1) = 0.39601, (iii) P(X <= 3) = 0.9879
2.5-2 (c) Bernoulli distribution (ii) mean = 0.55, variance = 0.2475 (iii) 0.55 (d) (ii) mean = 2.1, variance = 0.89 (iii) 0.7 (f) (ii) mean = 5.5, variance = 8.25 (iii) 0.2
2.5-6 0.99^99 = 0.36973
2.5-16 1 - 0.99^50 = 0.39499
2.5-18 e^(5t)
2.5-22 0.147
2.6-2 P(X=2) = 0.22404
2.6-6 e^(-0.5) = 0.60653
2.6-8 (a) 0.040091 (or 0.040428)
3.3-8 (a) c = 1
3.3-16 (b) F(x) = 0 (x <= 0), x/2 (0 < x < 1), 1/2 (1 <= x <= 2), x/2 - 1/2 (2 < x < 3), 1 (x >= 3) (c) pi_0.25 = 0.5 (d) pi_0.75 = 2.5
3.4-8 (a) X ~ exponential distribution with ( theta = 3/2), (b) 0.2636
3.4-18 P(No flaws in [0, 40']) = 0.30119
3.4-22: 0.75
3.5-8 (a) Gamma(7, 1/16), (b) 0.68663
3.5-10 a = 5.226, b = 21.03