STOKASTIK Sannolikhetsteori och statistikteori med tillämpningar
STOKASTIK Sannolikhetsteori och statistikteori med tillämpningar
STOKASTIK Sannolikhetsteori och statistikteori med tillämpningar
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2007-10-08 – sida 177 – # 181<br />
Svar till övningarna<br />
Kapitel 2<br />
2.1 a) Ω = Z = {. . . , −2, −1, 0, 1, 2, . . .}, A = N = {1, 2, . . .} <strong>och</strong> B =<br />
{1401, 1402, . . .}.<br />
b)A ∪ B = {1, 2, . . .} ∪ {1400, 1401, . . .} = {1, 2, . . .} = A <strong>och</strong> A ∩ B =<br />
{1, 2, . . .} ∩ {1400, 1401, . . .} = {1401, 1402, . . .} = B.<br />
c) Ac = {. . . , −2, −1, 0}, dvs ett negativt eller nollresultat, <strong>och</strong> A \ B =<br />
2.2<br />
{1, 2, . . . 1400}.<br />
a) Ω = {1, 2, . . .}, A = {1, 2, . . . , 10}, <strong>och</strong> B = {2, 4, 6, . . .}.<br />
b) A ∪ B = {1, 2, . . . 10, 12, 14, . . .}, A ∩ B = {2, 4, 6, 8, 10}.<br />
c) Ac = {11, 12, . . .} <strong>och</strong> A \ B = {1, 3, 5, 7, 9}.<br />
2.3 a) Ω = {1/1, 2/2, . . . , 31/1, 1/2, . . . , 31/12}, b) S = {1/9, . . . 30/9},<br />
O = {1/10, . . . 31/10}, c) V = {24/9, . . . 30/9, 1/10, . . . 23/10},<br />
2.4<br />
d) S \ V = {1/9, . . . 23/9}, O ∩ V = {1/10, . . . , 23/10} respektive S ∪ V =<br />
{1/9, . . . 30/9, 1/10, . . . 23/10}.<br />
A = {2, 4, 6}, P (A) = 3/6 = 1/2, B = {3, 6}, P (B) = 2/6 = 1/3,<br />
A ∩ B = {6}, P (A ∩ B) = 1/6, A ∪ B = {2, 3, 4, 6}, P (A ∪ B) = 4/6 = 2/3.<br />
2.5 P (A ∩ B) = 0.3.<br />
2.6 P (vinst) = P (H) + P (N) + P (T ) = 0.111.<br />
2.11 a) c = 6/π2 , b) P (B) = 49<br />
6π2 ≈ 0.827, c) P (A c<br />
1 ) = 1 − 6/π2 2.13<br />
≈ 0.392.<br />
P (A ∪ B ∪ C) = P (A) + P (B) + P (C) − P (A ∩ B) − P (A ∩ C) − P (B ∩ C) +<br />
2.14<br />
P (A ∩ B ∩ C).<br />
a) P (A ∪ B ∪ C) ≤ P (A) + P (B) + P (C),<br />
<br />
n<br />
b) P<br />
<br />
n<br />
≤ P (Ak).<br />
2.15 Sannolikheten att det blir Marie är 1/31 ≈ 0.03226, <strong>och</strong> sannolikheten att<br />
det blir en flicka är 17/31 ≈ 0.5484.<br />
2.16 a) Frekventistisk, b) subjektiv, c) frekventistisk.<br />
k=1<br />
Ak<br />
k=1