Теория вероятностей и математическая статистика. Билялов Р.Ф. - 41 стр.

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x
i
\ y
j
−1
X Y.
X Y ? P (X = 2, Y = 0)
P (X > Y ).
P (X = 1) = P (X = 1, Y = 0) + P (X = 1, Y = −1) =
(X = 1, Y = −1) = 0.8,
x
i
p
i
y
j
−1
p
j
P (X = 1, Y = −1) = 0.15 6=
P (X = 1)P (Y = −1) = 0.16. P (X = 2, Y = 0) = 0.05,
P (X > Y ) = 1 − 0.35 = 0.65.
ξ
p
ξ
(x) = αe
−αx
(x > 0).
η
1
=
√
ξ; η
2
= ξ
2
;
η
3
=
1
α
ln ξ; η
4
= 1 − e
−αξ
.
F
ξ
(x) =
x
R
0
p(t)dt = 1 − e
−αx
.
F
η
1
(x) = P (
p
ξ < x) = P (ξ < x
2
) = F
ξ
(x
2
), p
η
1
(x) = 2xαe
−αx
2
, (0 < x),
F
η
2
(x) = P (ξ
2
< x) = P (ξ <
√
x) = F
ξ
(
√
x), p
η
2
(x) =
α
2
√
x
e
−α
√
x
,
(0 < x),
F
η
3
(x) = P (
1
α
ln ξ < x) = P(ξ < e
αx
) = F
ξ
(e
αx
),
p
η
3
(x) = α
2
e
α(x−e
αx
)
, (−∞ < x < ∞),
F
η
4
(x) = P (1 − e
−αξ
< x) = P (e
−αξ
> 1 − x) = P (ξ < −
ln(1 − x)
α
) =
= F
ξ
(−
ln(1 − x)
α
), p
η
4
(x) = 1, 0 ≤ x ≤ 1.
 xi \ yj   −1     0      1    à) Íàéòè áåçóñëîâíûå çàêîíû ðàñïðå-
    1    0.15   0.3 0.35      äåëåíèÿ îòäåëüíûõ êîìïîíåíò X è Y.
    2    0.05 0.05     0.1    á) Óñòàíîâèòü, çàâèñèìû èëè íåò
êîìïîíåíòû X è Y ? â) Âû÷èñëèòü âåðîÿòíîñòè P (X = 2, Y = 0) è
P (X > Y ).
    Ðåøåíèå. à) P (X = 1) = P (X = 1, Y = 0) + P (X = 1, Y = −1) =
(X = 1, Y = −1) = 0.8, àíàëîãè÷íûì îáðàçîì ïîäñ÷èòûâàåì îñòàëü-
íûå âåðîÿòíîñòè è ïðèõîäèì ê ñëåäóþùèì ðÿäàì ðàñïðåäåëåíèé:
              xi       1      2                  yj   −1       0      1
              pi     0.8    0.2                  pj   0.2   0.35   0.45

á) çàâèñèìû, òàê êàê, íàïðèìåð, P (X = 1, Y = −1) = 0.15 6=
P (X = 1)P (Y = −1) = 0.16. â) P (X = 2, Y = 0) = 0.05,
P (X > Y ) = 1 − 0.35 = 0.65.
    Çàäà÷à 5.3. Ñëó÷àéíàÿ âåëè÷èíà ξ èìååò ïîêàçàòåëüíîå ðàñïðå-
äåëåíèå ñ ïëîòíîñòüþ ðàñïðåäåëåíèÿ pξ (x) = αe−αx (x  √ > 0). Íàéòè
ïëîòíîñòè ðàñïðåäåëåíèÿ ñëó÷àéíûõ âåëè÷èí: à) η1 = ξ; á) η2 = ξ 2 ;
â) η3 = α1 ln ξ; ã) η4 = 1 − e−αξ .
                          Rx
    Ðåøåíèå. Fξ (x) = p(t)dt = 1 − e−αx .
                            0
             p                                                  2
Fη1 (x) = P ( ξ < x) = P (ξ < x2 ) = Fξ (x2 ), pη1 (x) = 2xαe−αx , (0 < x),
                                    √         √              α    √
  Fη2 (x) = P (ξ 2 < x) = P (ξ <     x) = Fξ ( x), pη2 (x) = √ e−α x ,
                                                            2 x
                                    (0 < x),
                            1
           Fη3 (x) = P (      ln ξ < x) = P (ξ < eαx ) = Fξ (eαx ),
                            α
                                         αx )
                   pη3 (x) = α2 eα(x−e          , (−∞ < x < ∞),
                                                                     ln(1 − x)
Fη4 (x) = P (1 − e−αξ < x) = P (e−αξ > 1 − x) = P (ξ < −                       )=
                                                                         α
                           ln(1 − x)
               = Fξ (−               ), pη4 (x) = 1, 0 ≤ x ≤ 1.
                               α



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