Kurs_vysshei_matematiki_UP_Berkov_N.A._2007-2
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p
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ξ
p
M (ξ) = 1, 9, M (ξ2) = 7, 3 p1, p2, p3
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p1 = 0, 1 x2 |
x1 < x2 |
M (ξ) |
D(ξ) |
M (ξ) = 3, 9, D(ξ) = 0, 09 |
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M (ξi) = 1 · p + 0 · (1 − p) = p,
D(ξi) = M (ξi2) − M (ξi) 2 = 12 · p + 02 · (1 − p) − p2 = p − p2 = p(1 − p) = p · q.
M (ξ) = M (ξ1) + . . . + M (ξn) = n · p,
D(ξ) = D(ξ1) + . . . + D(ξn) = n · p · q.
ξ
M (ξ) = np; D(ξ) = npq.
n = 4 p = 0, 5
P4(0) = 0, 54 ≈ 0, 06; P4(1) = 4 · 0, 5 · 0, 53 ≈ 0, 25; P4(2) = C42 · 0, 52 · 0, 52 ≈ 0, 38; P4(3) = p4(1) ≈ 0, 25;
P4(4) = p4(0) ≈ 0, 06.
ξ
p
M (ξ) = n · p = 4 · 0, 5 = 2; D(ξ) = nqp = 4 · 0, 5 · 0, 5 = 1. M (ξ) = 2, D(ξ) = 1
n → ∞, p → 0 |
np → a |
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Pn(m) |
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am |
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P {ξ = m} = |
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e−a, a > 0. |
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m! |
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M (ξ) = k · KL .
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p(1 − p)3 |
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ξ A A p = p(A)
M (ξ) = |
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1 − p |
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ϕ(x) = |
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ϕ(x)dx = 1 = |
Cdx = 1 = |
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M (ξ) = 2; D(ξ) = |
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M (ξ) = 2; D(ξ) = 34
ϕ(x) = |
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ϕ(t)dt = |
λe−λtdt = −e−λt |
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M (ξ) = |
xϕ(x)dx = |
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xλe−λxdt = λ |
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xe−λxdx = |
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F(x) |
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du = dx |
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D(ξ) = |
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M (ξ) = |
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λ |
ξ1, ξ2, ξ3, . . .
λ
t a = λt
ξ F (x)
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F (x) = 0 |
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1 < x 2 F (x) = P (ξ < x) = P (ξ = 1) = 1/6. |
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ξ < 2 2 ξ < 3 |
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F (x) = P (ξ < x) = P {(ξ < 2) + (2 ξ < 3)} = |
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= P (ξ < 2) + P (2 ξ < 3). |
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P (ξ < 2) = 1/6 P (2 ξ < 3) |
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2 x < 3 |
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2 < x 3 F (x) = 1/6 + 1/6 = 1/3 |
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3 < x 4 |
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F (x) = P (ξ < x) = P (ξ < 3) + P (3 ξ < 4) = 1/3 + 1/6 = 1/2.
4 < x 5
F (x) = P (ξ < x) = P (ξ < x) + P (4 ξ < 5) = 1/2 + 1/6 = 2/3,
5 < x 6 F (x) = 2/3 + 1/6 = 5/6. x 6
F (x) = P (ξ < x) = P (ξ < 6) + P (6 ξ < x) = 5/6 + 1/6 = 1.