Kurs_vysshei_matematiki_UP_Berkov_N.A._2007-2
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0, 1, 2, . . . , 10 |
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(a; b) |
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(0, 3; 3) |
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ξ |
ζ |
η |
θ |
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x y z |
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ξ |
x1 |
x2 |
. . . |
xn |
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p |
p1 |
p2 |
. . . |
pn |
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ξ |
n |
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x1, . . . , xn, |
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pi = P {ξ = xi} |
ξ |
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n |
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ξ = x1, ξ = x2, . . . , ξ = xn |
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p1 + . . . + pn = 1.
ξ
ξ
p
ξ
ξ
p
0, 1 + p2 + 0, 3 + 0, 2 = 1 |
p2 = 0, 4 |
p2 = 0, 4
xi 
pi 
Pi |
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0,4 |
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0,3 |
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0,2 |
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0,1 |
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1 |
2 |
5 |
10 |
xi |
M 
ξ
ξ C
p
M (C) = C · 1 = C.
M (C · ξ) = C · M (ξ).
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ξ |
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Cξ |
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Cξ |
Cx1 |
Cx2 |
. . . |
Cxn |
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p |
p1 |
p2 |
. . . |
pn |
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M (Cξ) = Cx1p1 + Cx2p2 + . . . + Cxnpn = C(x1p1 + . . . + xnpn) = CM (ξ).
M (ξ + ζ) = M (ξ) + M (ζ).
ξ1, . . . , ξn 
C1, . . . , Cn 
M (C1ξ1 + . . . + Cnξn) = C1M (ξ1) + . . . + CnM (ξn).
M (ξ − ζ) = M (ξ) − M (ζ)
ξ 
ζ 
D(ξ) 0
D(C) = 0
D(C) = M (C − M (C))2 = M (C − C)2 = M (0) = 0
D(C · ξ) = C2 · D(ξ)
D(C · ξ) = M (C · ξ − M (C · ξ))2 = M (C · ξ − C · M (ξ))2 =
=M C · (ξ − M (ξ)) 2 = M C2 · (ξ − M (ξ))2 =
=C2 · M (ξ − M (ξ))2 = C2 · D(ξ).
ξ 
ζ 
D(ξ + ζ) = D(ξ) + D(ζ)
D(ξ −
−ζ) = D(ξ) + D(ζ)
D(ξ − ζ) = D ξ + (−1) · ξ = D(ξ) + (−1)2 · D(ξ) = D(ξ) + D(ζ).
M (ξ) 
D(ξ) 
ξ
σ(ξ) = D(ξ).
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σ(ξ) |
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D(ξ) = |
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= σ2(ξ) |
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D(ξ) = 9, 84 |
√ |
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≈ 3, 14 |
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σ(ξ) = |
9, 84 |
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σ(ξ) 0 |
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σ(C) = 0 |
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σ(Cξ) = |C| · σ(ξ) |
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σ(ξ) |
ξ ζ σ(ξ+ζ) = |
σ2(ξ) + σ2(ζ) |
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n = 10, q = 0, 98, p = 1 − q = 0, 02, m = 2
P10(2) = C102 · p2 · q8 = 2!8!10! · 0, 022 · 0, 988 ≈ 0, 015.
P10(2) ≈ 0, 02
m = 0, 1, 2, 3, 4, 5
n = 5 m = 0 p = 0, 9 q =
0, 1 
P5(0) = C50 · (0, 9)0 · (0, 1)5 = 10−5.
m
P5(1) = C51 · (0, 9)1 · (0, 1)4 = 1!4!5! · 0, 9 · 10−4 ≈ 0, 0005,
P5(2) = C52 · (0, 9)2 · (0, 1)3 = 2!3!5! · 0, 81 · 10−3 ≈ 0, 0081,
P5(3) = C53 · (0, 9)3 · (0, 1)2 = 3!2!5! · 0, 729 · 10−2 ≈ 0, 0729,
P5(4) = C54 · (0, 9)4 · (0, 1)1 = 4!1!5! · 0, 6561 · 0, 1 ≈ 0, 32805,
P5(5) = C55 · (0, 9)5 · (0, 1)0 ≈ 0, 59049.
5
P5(m) = 1.
m=0
p = 0, 9
7%
n = 6, p = 0, 07, q = 0, 93.
P1 
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m = 0, P6(0) = C60 · (0, 07)0 · (0, 93)6 ≈ 0, 647. |
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P6(1) |
P6(2) |
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P6(1) = C61 · (0, 07)1 · (0, 93)5 ≈ 0, 292, |
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P6(2) = C62 · (0, 07)2 · (0, 93)4 ≈ 0, 055. |
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P2 |
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P2 = P6(0) + P6(1) + P6(2) ≈ 0, 994. |
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6 |
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P6(m) = 1, |
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m=0 |
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P3 |
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6 |
6 |
2 |
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P3 = |
P6(m) = P6(m) − |
P6(m) ≈ 1 − 0, 994 = 0, 006. |
m=3 |
m=0 |
m=0 |
P1 ≈ 0, 647 P2 ≈ 0, 994 P3 ≈ 0, 006.
n = 50, |
p1 = 0, 92, |
p2 = 0, 06, |
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p3 = 0, 02, |
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k2 = 3, k3 = 1, k1 = n − k2 − k3 = 46. |
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P50(46, 3, 1) = |
50! |
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· 0, 9246 · 0, 063 · 0, 021 = |
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46!3!1! |
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47 · 48 · 49 · 50 |
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0, 02162 |
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0, 000216 |
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0, 02 |
≈ |
0, 086. |
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6 |
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0, 9246 |
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m 
A 
n 
n = 8, p = 0, 6, q = 0, 4
(n + 1)p − 1 m (n + 1)p |
4, 4 m 5, 4. |
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m = 5. |
P8(5) = C85 · (0, 6)5 · (0, 4)3 = |
8! |
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· 0, 07776 · 0, 064 ≈ 0, 279. |
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3!5! |
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m = 5, P8(5) ≈ 0, 28