Ëàáîðàòîðíàÿ ðàáîòà: Òåîðèÿ âåðîÿòíîñòåé
Ì (Õ+Ó)
2 Õ , Õ |
0 | 1 |
Ð | 0,3 | 0,7 |
2 Õ , Õ |
0 | 1 |
Ð | 0,6 | 0,4 |
2
Ì (Õ) = 0,7 = Ì (Õ )
2
Ì (Ó) = 0,4 = Ì ( Ó )
= 0,7 + 2 * 0,7 * 0,4 + 0,4 = 1,66
16. Õ, Ó íåçàâèñèìûå íåèçâåñòíûå âåëè÷èíû Ïðèíèìàþò çíà÷åíèå 0 è 1.
(çàäàíèå êàê â 15).
Õ | 0 | 1 |
Ð | 0,3 | 0,7 |
Ó | 0 | 1 |
Ð | 0,5 | 0,5 |
õ - ó
Ì (3 ) - ?
õ-ó õ -ó õ -ó
Ì (3 ) = Ì (3 * 3 ) =Ì (3 ) * Ì (3 ) = 2,4 * 2 = 1,6
3
õ 3 |
1 | 3 |
Ð | 0,3 | 0,7 |
-ó 3 |
1 |
1 3 |
Ð | 0,5 | 0,5 |
Õ -ó
Ì (3 ) = 0,3 + 2,1 = 2,4 Ì (3 ) = 0,5 + 0,5 = 4 * 0,5 = 1
3 3 3
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17. Ïðîèçâîäèòñÿ 10240 íåçàâèñèìûõ èñïûòàíèé, ñîñòîÿùèõ â òîì, ÷òî
ïîäáðàñûâàþòñÿ 9 ìîíåò
Õ – ÷èñëî èñïûòàíèé, â êîòîðûõ âûïàëî 3 ãåðáà
Ì (Õ) -?
1-èñïò. - 9 ìîíåò
9 èñïûòàíèé Ð = 1
2
3 3 6 3 9
Ð(Ã = 3) = Ñ9 * ( 1 ) * ( 1 ) = Ñ9 * ( 1 ) = 84 * 1 - 21 = …
2 2 2 512 128
n = 10240 èñïûòàíèé
Ð = 21 ; Ì (Õ) = np = 21 * 10240 = 1680
128 128
18. Â ñåðèè íåçàâèñèìûõ èñïûòàíèé (îäíî èñïûòàíèå çà åä.âðåìåíè)
âåðîÿòíîñòü íàñòóïëåíèÿ À ðàâíà 1
8.
Ïóñòü Ò-âðåìÿ îæèäàíèÿ íàñòóïëåíèÿ ñîáûòèÿ À 14 ðàç. Íàéòè Ì (Ò)1 Ä (Ò).
Õ1 – âðåìÿ îæèäàíèÿ äî ïåðâîãî íàñòóïëåíèÿ À
Õ2 – âðåìÿ îæèäàíèÿ îò ïåðâîãî íàñòóïëåíèÿ À äî 2-ãî
Ò = Õ1 + Õ2 +Õ3 + ..Õ14
Õi Ð = 1
8 7/8
Ì (Õi) = 1 = 8 ; d = 7 Ä (Õi) = d = = 56
8 8 2 2
p 1/8
Ì (Ò) = 14Ì * (Õ1) 14 * 8 = 112
Ä (Ò) = Ä(X1 ) = 14 * 56 = 784
19. Âåëè÷èíû Õ1 …..Õ320 ðàñïðåäåëåíû ïî Áèíîìèíàëüíîìó çàêîíó ñ ïàðàìåòðàìè
ï =4, ð = 3 Íàéòè Ì (Õ1 + Õ2 + …+ Õ320)=?
8
2 2 2
Ì (Õ1 + …..+Õ 320) = 320Ì (Õ1 ) = Õ1 – áèíîìèíàëüíîå
2 2 Ì (Õ1) = ïð = 3
= Ì(Õ1 ) = Ä(Õ1) + Ì (Õ1) = 2
2 Ä (Õ1 ) = nðq = 3 * 5 = 5
= 15 + 3 = 15 + 9 = 51 2 8 16
16 2 16 4 16
= 320 * 51 = 1020
16
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20. Âåëè÷èíû Õ1 …..Õ18 ðàñïðåäåëåíû ïî çàêîíó Ïóàññîíà ñ îäèíàêîâûì
ìàò. îæèäàíèÿì ðàâíûì 8.
2 2
Íàéòè Ì (Õ1 +…+ Õ18 ) - ?
M (Õ) = Ä (Õ) = l = 8
2 2 2 2
Ì (Õ1 +…+ Õ18 ) = 18 Ì (Õ1 ) = 18 (Ä (Õ1) + Ì (Õi ) )=18(8 + 64)=18 * 72=1296
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21. Õ ðàâíîìåðíî ðàñïðåäåë¸í íà îòð. [ - 8,2 ]
Ð ( 1 )>5 = Ð (0< Õ <1 ) = > (0< Õ <0,5) =
Õ 5
1 – 5 >0 ; 1 – 5Õ > 0; Õ –1/5 < 0 Û (0< Õ <0,5)
Õ Õ Õ
1 5Õ > 0; Õ – 1/5 < 0
Õ Õ
[ õ, â ]
0,Õ>à 0; Õ <à
f (Õ)= 1 ; à < Õ < â F (Õ) = õ à ; à £ Õ £ à Û 0< Õ 1/5
â –î â –à
0,Õ > â 1, Õ >B
F (Õ) = Õ + 8 = F (1/5) - F ( 0 ) =1/5 + 8 - 8 = 1
5 10 10 50
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22. Õ ðàâíîìåðíî ðàñïðåäåëåíà íà îòð. [ -17; 10 ]
2 2
Ð ( Õ > 64) = 1- Ð ( Õ < 64) = 1 – 16
27
2
Ð (Õ < 64 ) = Ð (-8 < Õ <8) =
0; Õ < -17
F(Õ) = Õ + 17 , -17 £ Õ £ 10
27
1, Õ > 10
= F (8) – F (-8) = 8 + 17 - -8 + 17 = 16
27 27 27
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23. Õ – ðàâíîìåðíî ðàñïðåäåëåíà íà îòð. [ -1; 1 ]
8/9 X [a,b] ; f (x)
Ì ( Õ ) a 0; x <-1
M(x)= ∫ x f(x) dx f (x)= -1<x<1
b 0; x>1
a
M(y(x))=∫ y (x) f (x) dx
b
8/9 1 8/9 17/9 1
M(X ) = ∫ ½* X DX = ½ * X = 9/17
-1 17/9 -1
24. Õ – ðàâíîìåðíî ðàñïðåäåëåíà íà îòð. [ 0.1 ]
9/10 9/10
Ä ( 19Õ ) = 361 (Õ )
9/10 9/10 2 2 9/10 9/4 2 9/10 9/10 * 2
Ä (Õ ) = Ì ( (Õ ) ) - Ì (Õ ) = Ì (Õ ) - Ì (Õ ) Õ
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25. Õ – ðàâíîìåðíî ðàñïðåäåëåíà íà îòð. [ 5; 8 ] * Ä (24x+ 36) - ?
Ä (24Õ + 36) = Ä (24Õ) = 576 * Ä (Õ) = 576 * 3 = 432
2 4
Ä (Õ) = ( â – à )
12
2
Ä (Õ) = 8 – 5 = 9 = 3
12 12 4
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26. Õ1,……Õ2 – Íåçàâèñèìûå è ðàñïðåäåëåííûå ïî ïîêàçàòåëüíîìó çàêîíó.
2
Íàéòè Ì [ (Õ1 + Õ2 + …..+ Õ10) ], åñëè Ì (Õi ) = 4.
Ì (Õ) = 1
l
Ä (Õ) = 1
2
l
M (Õi ) = > Ä (Õi) = 16
2 2 2
Ì [ (Õ1 +….+ Õ10) ]=Ä(Õ1 +…+ Õ10) + Ì (Õ1 +….+ Õ10) =10Ä (Õ1)+[ 10Ì (Õ1) ]=
2
= 160 + ( 10 * 4) = 1760
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2
Ì(Õ) =1/ l ; Ä(Õ) = 1/l
27. Õ –ðàñïðåäåëåí ïî ïîêàçàòåëüíîìó ïðèçíàêó
2
Íàéòè Ì [ (Õ + 8) ] , åñëè Ä (Õ) = 36 Ì (Õ)=6
2 2 2 2
Ì (Õ + 8) = M(Õ + 16õ + 64) = Ì (Õ ) + 16Ì (Õ) + Ì (64) = Ä (Õ) + Ì (Õ) +
+ 16 Ì(Õ) + 64 =36 + 36 + 96 + 64 =232
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28. Õ ïîêàçàòåëüíîå ðàñïðåäåëåíèå; Õ – ïîêàçàòåëüíûé çàêîí
0, Õ < 0
F (Õ) = -2õ
1 – å , Õ >0, Íàéòè Ln (1 Ð ( Õ < 6) ) = Ln (1 – F (6) ) =
-6/7 -6/7 -6/7
= F (6) = 1 – å = Ln ( 1 – (1 – å ) ) = Ln å = - 6/7
29. (Õ) - ñëó÷àéíàÿ âåëè÷èíà
0, Õ < 10
ƒ (Õ) = Ñ ; Õ ≥ 10
5
Õ
Ñ - ? ; Ì (Õ) - ?
¥ ¥ îïð. B ¥ -5
∫ ƒ (Õ)dõ = 1 => ∫ ñ dõ = lim ∫ = cdx = C lim ∫ X dx =
10 10 5 b->¥ 10 5 b->¥ 10
Õ X
b
-4 -4 4 4 4
= C * lim X = C lim - b + 10 = C * 10 = > 1 = C 10 = >
b->¥ -4 b->¥ 4 4 4 4
10
4
=> C = 4 * 10
0; Õ < 10
ƒ (Õ) = 4
4 * 10 , Õ ³ 10
5
Õ
¥ ¥ 4
Ì (Õ) = ∫ Õ ƒ (Õ) dx = ∫ 4 * 10 dx
10 10 4
Õ
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30. Õ – íîðìàëüíàÿ ñëó÷àéíàÿ âåëè÷èíà
Ì (Õ) = 16
Ä (Õ) = 25
? Ð (Õ>10,5)
= 1 - f 10,5 – 16 = 0,5 + f (1,1) = 0,5 + 0,364 = 0,864
2 5
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1. Ð (d £ X £ b ) = f b – m - f d - m
d d
2. P ( X < b ) = 1 + f b – m
2 d
3. P ( X > b ) = 1 - f b – m
2 d