Теория вероятностей и математическая статистика. Билялов Р.Ф. - 44 стр.

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P (A) = e
−(λ
1
+λ
2
+λ
3
)
(1 + λ
1
+ λ
2
+ λ
3
), λ
1
= 0.3,
λ
2
= 1, λ
3
= 0.7. P (A) = 3e
−2
= 0.406. P (
¯
A) = 0.594.
(θ = k), k = 1, 2, 3, 4; (ν = l), l = 1, 2, 3, . . . .
(θ, ν) :
ν = 1
ν = 2 θ = 1 θ = 2 θ = 3 θ = 4
X
N(m, σ
2
).
p
k
kσ
k = 1, 2, 3).
p
1
≈ 0.683, p
2
≈ 0.954, p
3
≈ 0.997.
X
N(m, σ
2
). p
1
= P (X ≥ x
n2
) p
2
= P (x
n1
≤ X ≤ x
n2
),
x
n1
x
n2
p
1
= 0.1587, p
2
= 0.68.
F
X,Y
(x, y),
m
X
m
Y
.
x \ y y ≤ −1 −1 < y ≤ 0 0 < y ≤ 1 1 < 0
x
≤
1 0 0 0 0
1 < x ≤ 2 0 0.15 0.45 0.8
2 < x 0 0.2 0.55 1
m
X
= 1.2, m
Y
= 0.25
ξ
a = 1, σ
2
= 1.
η
1
= ξ
2
; η
2
= e
ξ
ìàøèíà ðàáîòàåò. P (A) = e−(λ1 +λ2 +λ3 ) (1 + λ1 + λ2 + λ3 ), λ1 = 0.3,
λ2 = 1, λ3 = 0.7. P (A) = 3e−2 = 0.406. P (Ā) = 0.594.
    Çàäà÷à 5.23. Èñïîëüçóÿ òàáëèöó ñëó÷àéíûõ ÷èñåë, âûïèñàòü 10
ðåàëèçàöèé èñïûòàíèé, îïèñàííûõ â çàäà÷å 5.17. Íàéòè ÷àñòîòû ñî-
áûòèé: (θ = k), k = 1, 2, 3, 4; (ν = l), l = 1, 2, 3, . . . .
    Ðåøåíèå.  òàáëèöå ñëó÷àéíûõ ÷èñåë îïóñêàåì ÷èñëà 0, 7, 8, 9.
Ïîëó÷àåì ñëåäóþùèå ðåàëèçàöèè ñëó÷àéíîãî âåêòîðà (θ, ν) : (1,1),
(1,3), (1,2), (2,3), (1,3), (2,2), (1, 1), (1,3), (2,3), (1,4). ×àñòîòû: ν = 1
 0.7; ν = 2  0.3; θ = 1  0.2; θ = 2  0.2; θ = 3  0.5; θ = 4  0.1.

                    Çàäà÷è äîìàøíåãî çàäàíèÿ.
      Çàäà÷à 14.362. Ñëó÷àéíàÿ âåëè÷èíà X ðàñïðåäåëåíà ïî çà-
êîíó N (m, σ 2 ). Ïîëüçóÿñü òàáëèöåé ôóíêöèè íîðìàëüíîãî ðàñïðåäå-
ëåíèÿ, âû÷èñëèòü âåðîÿòíîñòü pk òîãî, ÷òî îòêëîíåíèå âåëè÷èíû Õ
îò åå ìàòåìàòè÷åñêîãî îæèäàíèÿ íå ïðåâçîéäåò âåëè÷èíû kσ (îòâåò
ïîëó÷èòü äëÿ òðåõ çíà÷åíèé k = 1, 2, 3).
                              Îòâåò: p1 ≈ 0.683, p2 ≈ 0.954, p3 ≈ 0.997.
   Çàäà÷à 14.373. Ñëó÷àéíàÿ âåëè÷èíà X ðàñïðåäåëåíà ïî çàêîíó
N (m, σ 2 ). Âû÷èñëèòü p1 = P (X ≥ xn2 ) è p2 = P (xn1 ≤ X ≤ xn2 ), ãäå
xn1 è xn2  òî÷êè ïåðåãèáà ïëîòíîñòè ðàñïðåäåëåíèÿ âåðîÿòíîñòåé.

                                            Îòâåò: p1 = 0.1587, p2 = 0.68.

   Çàäà÷à 14.379. (ïðîäîëæåíèå Çàäà÷è 14.378). Ïîñòðîèòü ôóíê-
öèþ ðàñïðåäåëåíèÿ FX,Y (x, y), îôîðìèâ ðåçóëüòàò â âèäå òàáëèöû,
è íàéòè mX è mY .
                   x\y  y ≤ −1 −1 < y ≤ 0 0 < y ≤ 1 1 < 0
                   x≤1     0       0          0        0
           Îòâåò:
                  1

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