From 82629bcab972c4a06fff257964a00c96c87eaf99 Mon Sep 17 00:00:00 2001 From: nik Date: Mon, 27 Oct 2025 21:31:36 +0300 Subject: [PATCH] add prac 8 --- pracs/prac8/ans.pdf | 2270 ++++++++++++++++++++++++++++++++++++++++++ pracs/prac8/ans.typ | 174 ++++ pracs/prac8/task.pdf | Bin 0 -> 733816 bytes 3 files changed, 2444 insertions(+) create mode 100644 pracs/prac8/ans.pdf create mode 100644 pracs/prac8/ans.typ create mode 100644 pracs/prac8/task.pdf diff --git a/pracs/prac8/ans.pdf b/pracs/prac8/ans.pdf new file mode 100644 index 0000000..274f8d6 --- /dev/null +++ b/pracs/prac8/ans.pdf @@ -0,0 +1,2270 @@ +%PDF-1.7 +% + +1 0 obj +<< + /Type /Pages + /Count 3 + /Kids [146 0 R 148 0 R 150 0 R] +>> +endobj + +2 0 obj +<< + /Type /Outlines + /First 3 0 R + /Last 3 0 R + /Count 1 +>> +endobj + +3 0 obj +<< + /Parent 2 0 R + /Title + /Dest 145 0 R +>> +endobj + +4 0 obj +<< + /Type /StructTreeRoot + /RoleMap << + /Datetime /Span + /Terms /Part + /Title /P + /Strong /Span + /Em /Span + >> + /K [8 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Случайная величина $xi$ задана функцией распределения $F_xi (x) = cases(0 ", " x lt.eq 0, a sin 2 x ", " 0 lt x lt.eq pi/2, 1 ", " x gt pi/2)$ коэффициент $a$, плотность распределения. + +*Решение*: + +Найдем $a$: + +Чтобы была правостоянтной и непрерывной в точке $x = pi/2$: + +$ +a sin^2(pi/2) = a = 1 arrow.double a = 1. +$ + +Найдем плотность $f_xi (x)$ + +$ +f_xi (x) = F'(x) = cases(0 ", " x lt.eq 0, frac(d, d x) sin^2 x = 2 sin x cos x = sin(2 x) ", " 0 lt x lt pi/2, 0 ", " x lt.eq pi/2.) +$ + +*Ответ*: $a = 1, space f_xi (x) = cases(sin(2x) ", " 0 lt x lt pi/2, 0 ", иначе.")$ + +#line(length: 100%) + +*2*. Плотность распределения СВ $xi$ равна $f_xi (x) = cases(0 ", " x lt.eq 0 ", " x gt pi/4, a cos 2 x ", " 0 lt x lt.eq pi/4, 1 ", " x gt pi/2)$. Найти коэффициент $a$, функцию распределения и вероятность попадания в интервал $(0, pi/6)$. + +*Решение*: + +Коэффициент $a$: +$ +integral_0^(pi/4) a cos 2 x space d x = 1 arrow.double a dot frac(sin(pi/2), 2) = a/2 = 1 arrow.double a=2 +$ + +Функция распределения $F_xi (x)$: + +$ +F_xi (x) = cases(0 ", " x lt.eq 0, integral_0^x 2 cos 2 t space d t = sin 2 x ", " 0 lt x lt.eq pi/4, 1 ", " x gt pi/4) +$ + +Вероятность $P(0 lt xi lt pi/6)$: +$ +F_xi (pi/6) - F_xi (0) = sin(2 dot pi/6) = sin(pi/3) = frac(sqrt(3), 2). +$ + +*Ответ*: $a = 2, space F_xi (x) = cases(0 ", " x lt.eq 0, integral_0^x 2 cos 2 t space d t = sin 2 x ", " 0 lt x lt.eq pi/4, 1 ", " x gt pi/4), space P(0 lt xi lt pi/6) = frac(sqrt(3), 2). +$ + +#line(length: 100%) + +*3*. Плотность распределения СВ $xi$ равна $f_xi (x) = cases(0 " , " x lt.eq -1 " , " x gt 1, frac(a, sqrt(1 - x^2)) " , " -1 lt x lt.eq 1)$. Найти коэффициент $a$, функцию распределения и вероятность попадания в интервал $(0, 1/2)$. + +*Решение*: + +Коэффициент $a$: + +$ +integral_(-1)^1 frac(a, sqrt(1 - x^2)) d x = a[arcsin x]_(-1)^1 = a pi = 1 arrow.double a = 1/pi +$ + +Функция распределения: +$ +F_xi (x) = cases(0 ", " x lt.eq -1, frac(arcsin x + pi/2, pi) ", " -1 lt x lt.eq 1, 1 ", " x gt 1) +$ + +Вероятность $P(0 < xi < 1/2)$: +$ +F_xi (1/2) - F_xi(0) = 2/3 - 1/2 = 1/6 +$ + +*Ответ*: $a = 1/pi, space F_\xi(x) = cases(0 ", " x lt.eq -1, frac(arcsin x + pi/2, pi) ", " -1 lt x lt.eq 1, 1 ", " x gt 1), space P(0 lt xi lt 1/2) = 1/6$. + +#line(length: 100%) + +*4*. Автобусы некоторого маршрута идут строго по расписанию. Интервал движения 5 мин. Найти вероятность того, что пассажир, подошедший к остановке, будет ожидать автобус менее 3 мин. + +*Решение*: + +Пусть $xi$ — время ожидания автобуса. + +Так как пассажир приходит случайно, $xi$ равномерно распределена на интервале от 0 до 5 минут (от последнего автобуса до следующего): + +$ +xi tilde U(0, 5) +$ + +Плотность равномерного распределения на $[0,5]$: + +$ +f_xi (x) = 1/5, space.quad 0 lt.eq x lt.eq 5 +$ + +Функция распределения: + +$ +F_xi (x) = P(xi lt.eq x) = integral_0^x f_xi (t) space d t = x/5, space.quad 0 lt.eq x lt.eq 5 +$ + + +Находим вероятность + +Нужно $P(xi lt 3)$: + +$ +P(xi lt 3) = integral_0^3 f_xi (x) space d x = integral_0^3 1/5 space d x = 3/5 = 0.6 +$ + +*Ответ*: $0.6$. + +#line(length: 100%) + +*5*. Случайная величина $xi$ задана плотностью $f_xi (x) = 2 e^(-2x)$. Найти вероятность попадания в интервал $(1, 2)$. + +*Ответ*: $P(1 < xi < 2) = integral_1^2 2 e^(-2 x) space d x = [-e^(-2x)]_1^2 = e^(-2) - e^(-4) approx 0.117$. + +#line(length: 100%) + +*6*. Производится взвешивание некоторого вещества. Случайные ошибки взвешивания подчинены нормальному закону распределения с $a = 0, space sigma = 20 "г."$ Найдите вероятность того, что взвешивание будет произведено с ошибкой, не превосходящей, по модулю $10 "г."$ + +*Решение*: Для нормальной случайной величины вероятность через стандартное нормальное распределение $Z tilde N(0,1)$ вычисляется так: + +$ +Z = frac(xi - a, sigma) +$ + +Здесь $a = 0, sigma = 20$, значит: + +$ +Z = frac(xi, 20) +$ + +Перевод интервала + +Нам нужно: + +$ +P(-10 lt.eq xi lt.eq 10) = P ((-10)(20) lt.eq Z lt.eq (10)(20)) = P(-0.5 lt.eq Z lt.eq 0.5) +$ + +Используем функцию стандартного нормального распределения $Phi(z)$ + +$ +P(-0.5 lt.eq Z lt.eq 0.5) = Phi(0.5) - Phi(-0.5) +$ + +Свойство функции распределения: + +$ +Phi(-z) = 1 - Phi(z) +$ + +Тогда: + +$ +P(-0.5 lt.eq Z lt.eq 0.5) = Phi(0.5) - (1 - Phi(0.5)) = 2 Phi(0.5) - 1 +$ + +Находим численно + +Стандартные таблицы: + +$ +Phi(0.5) approx 0.6915 +$ + +Тогда: + +$ +P(-0.5 lt.eq Z lt.eq 0.5) = 2 dot 0.6915 - 1 = 1.383 - 1 = 0.383 +$ + +*Ответ*: $0.383$. + diff --git a/pracs/prac8/task.pdf b/pracs/prac8/task.pdf new file mode 100644 index 0000000000000000000000000000000000000000..63ec0bba42cc9b578c8423a01efa8674aedca9c5 GIT binary patch literal 733816 zcmYg#V_;-Uv~?zSCblNFZQHgrv2EL&*qPY2I<{@w6MdQc?tSn3(|xM?RPEY(ueEkn zHHo~iC=CNG8#FXC0X>1O;ZJCC31br*CkrQc3TP7>xw)a~ zL@lhGOdJU4M6C>*OoUC0Y>iEz>7-3;%$&>#*q9g?+5c-X5HK;bF>TyAW!Ft1XnSbEcUlyK1b$okO($lE$8S)q_SA z6S!O!yT=0%YkVN)9r!)W25$p)#GVJIU_hC0@9nEt!?JD59tUa|e4Kf^QN@do;RrPJ z-=ujV^*h`TU*_8B*6pr>dpH=l(Q{g@$}CT`KA#!v`cr_(VUC<~9DNStXi@wfZ6V^8 za+tg&*wO^R*5`5zy>>Z{D%0-%Oe9bAu-JeC5XQ5dKZ46$w(7CAGavZz+D$LdPwIJZ z8>@i`TxVaW29YrXz)Wr#+q@m8^gQw^MD0N~Uj_X_5**FwboD!j7(rSd>Zd>Z?$9PP zKh_huW?t8i7@o6^r=^DRPFr;;@!HD&Fl)((iu z-b<+TvThz$KTdv-qZ_HC!pQkI_3=!@nnhectX4DszJTK|b2Gq4A#3UjHZy3T#Cj@R z6h+_O*`nog)h7<8T;IRJh!g(dulA|UnsSK3c(6bm8gi%@BB&0^bB!N#4huPw9SB9y zx;^dg&Jq*@SKh*V66<5#wgr;l>$#!hp?&_L)5jI0kkS|S8XqKVi{AMJNgXF&UQmf9e5|N3x8DLMYxAr1LRKPM5G5AiFDO77)9ql2uSTi 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