STOKASTIK Sannolikhetsteori och statistikteori med tillämpningar
STOKASTIK Sannolikhetsteori och statistikteori med tillämpningar
STOKASTIK Sannolikhetsteori och statistikteori med tillämpningar
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180 SVAR TILL ÖVNINGARNA<br />
2007-10-08 – sida 180 – # 184<br />
3.25 pZ(−1) = 0.995 <strong>och</strong> pZ(100) = 0.005. E(Z) = −0.495. V (Z) ≈ 50.75.<br />
3.26 pX(k) = 1/100, k = 1, . . . , 100, E(X) = 50.5, D(X) = √ 833.25 ≈ 28.87.<br />
3.27 a) E(X) = 1.8 <strong>och</strong> D(X) = 1.24, b) 0.4435, c) 0.2642, d) 0.0683.<br />
3.28 a) E(Y ) = 12 <strong>och</strong> D(Y ) = √ 3 ≈ 1.73, b) 0.4050, c) 0.9204.<br />
3.29 pX(3) ≈ 0.129, P (X ≥ 3) = 1 − FX(2) = 0.1841.<br />
3.31 a) 0.1480, b) 0.3277, c) 0.2749.<br />
3.32 a) E(X) = 1.2, D(X) = 0.748, b) 0.3.<br />
3.33 0.866.<br />
3.35 P (Y = 2) = 0.309 <strong>och</strong> P (X = 2) = 0.316, dvs ganska lika varandra.<br />
3.36 a) 0.353, b) 0.380, c) 1.32 respektive 1.15.<br />
3.37 a) 0.0859, b) 0.4240.<br />
3.38 1/ √ λ.<br />
3.39 0.083.<br />
3.40 a) 0.0782, b) 0.2517 (mer exakt värde =??).<br />
3.41 a) 0.1029, b) 0.343, c) 0.51, d) E(X)=3.33 <strong>och</strong> D(X)=2.789.<br />
3.42 a) 0.16, b) 0.8, c) 0.992, d) E(Y )=0.25 <strong>och</strong> D(X)=0.5590.<br />
3.43 a) 0.488, b) 4.<br />
3.44 R(X) = √ q, R(Y ) = 1/ √ q, <strong>och</strong> R(Z) = q/r.<br />
3.45 a) 0.5625, b) 0.0659, c) 0.9624, d) E(X) = 4, D(X) = 1.155.<br />
3.46 a) 0.1073, b) 0.1890, c) k = 12.<br />
3.48 a) 0.3 b) 0.8, c) E(U) = 15, D(U) = 10/ √ 12 ≈ 2.89.<br />
3.49 a) fU (u) = 1/10 för −5 ≤ u ≤ 5 <strong>och</strong> fU (u) = 0 för övriga u.<br />
b) FU (u) = 0, u < −5, FU (u) = (u + 5)/10 för −5 ≤ u ≤ 5 <strong>och</strong> FU (u) = 1,<br />
u > 5.<br />
c) λU (u) = 1/(5 − u) för −5 < u < 5 <strong>och</strong> λU (u) = 0 annars.<br />
3.50 a) 0.1, b) 0.1, c) 1/ √ 3 ≈ 0.577.<br />
3.52 a) 1 − e −2 ≈ 0.865, b) e −0.8 − e −1.6 ≈ 0.247, E(Y ) = 0.25 <strong>och</strong> V (Y ) =<br />
0.0625.<br />
3.53 a) e −0.4 − e −0.8 ≈ 0.221, b) e −0.8 − e −1.2 ≈ 0.148, c) R(T ) = 1.<br />
3.54 a) e −3 ≈ 0.0498, b) 1 − e −3/12 ≈ 0.221, 1/3 år.<br />
3.56 a) 0.9332, b) 0.0919, c) 0.3085.<br />
3.57 a) 0.3821, b) 0.7224, c) 0.8181.<br />
3.58 a) 0.6915, b) 0.4938, c) 0.0401.<br />
3.59 a) 0.0668, b) 0.9599, c) 0.4649.<br />
3.60 a) 0.8849, b) 0.7517, c) 0.2666.<br />
3.61 a) Φ(a/σ), b) 2Φ(a/σ) − 1, c) Φ(b/σ) + Φ(a/σ) − 1.<br />
3.62 a) 2Φ(a) − 1, b) Φ(b) − 0.5, c) Φ(b) − Φ(a).<br />
3.63 a) 0.47, b) 1.48, c) 2.05.<br />
3.64 a) 116.45, b) 119.6, c) 125.76.<br />
3.65 xα = µ + σλα.