Действительно, по той ссылке обман: вместо книги "Конспекты лекций по теории вероятности математической статистике и случайным процессам" дана книга "Конспекты лекций по теории вероятности и математической статистике". В указанной книге необходимый материал точно есть. Поищите.
В первой задаче требуется найти плотность вероятности величины
для каждого фиксированного
. Подобные задачи вы должны были решать на теории вероятности.
Вторая задача: даны произвольные натуральные числа
,
, по ним определяются случайные виличины
(координаты частицы в моменты времени). Зная случайные величины, можно найти их корреляционную функцию(опять же задачка по ТВ)
. Это функция от двух переменных, но в случаях, когда она зависит только от разности
, пишут
.
Третью задачу можно решить через теорему Винера-Хинчина.
![$% Предварительный просмотр исходного кода абзацев 0-2
\textbf{5.} \textcyr{\char205\char224\char233\char242\char232} \textcyr{\char238\char228\char237\char238\char236\char229\char240\char237\char243\char254}
\textcyr{\char239\char235\char238\char242\char237\char238\char241\char242\char252}
\textcyr{\char226\char229\char240\char238\char255\char242\char237\char238\char241\char242\char232}
\textcyr{\char241\char235\char243\char247\char224\char233\char237\char238\char227\char238}
\textcyr{\char239\char240\char238\char246\char229\char241\char241\char224}
$\xi(t)=\sin(\omega t+\varphi)$, \textcyr{\char229\char241\char235\char232}
\textcyr{\char241\char235\char243\char247\char224\char233\char237\char224\char255}
\textcyr{\char226\char229\char235\char232\char247\char232\char237\char224}
$\varphi$ \textcyr{\char240\char224\char241\char239\char240\char229\char228\char229\char235\char229\char237\char224}
\textcyr{\char240\char224\char226\char237\char238\char236\char229\char240\char237\char238}
\textcyr{\char237\char224} \textcyr{\char238\char242\char240\char229\char231\char234\char229}
$[-\pi,\pi]$, \textcyr{\char224} $\omega$ - \textcyr{\char228\char229\char242\char229\char240\char236\char232\char237\char232\char240\char238\char226\char224\char237\char237\char224\char255}
\textcyr{\char226\char229\char235\char232\char247\char232\char237\char224}.
\textbf{6.} \textcyr{\char194} \textcyr{\char237\char224\char247\char224\char235\char252\char237\char251\char233}
\textcyr{\char236\char238\char236\char229\char237\char242} \textcyr{\char226\char240\char229\char236\char229\char237\char232}
\textcyr{\char247\char224\char241\char242\char232\char246\char224}
\textcyr{\char237\char224\char245\char238\char228\char232\char242\char241\char255}
\textcyr{\char226} \textcyr{\char237\char224\char247\char224\char235\char229}
\textcyr{\char234\char238\char238\char240\char228\char232\char237\char224\char242}.
\textcyr{\char194} \textcyr{\char234\char224\char230\char228\char251\char233}
\textcyr{\char246\char229\char235\char238\char247\char232\char241\char235\char229\char237\char237\char251\char233}
\textcyr{\char236\char238\char236\char229\char237\char242} \textcyr{\char226\char240\char229\char236\char229\char237\char232}
\textcyr{\char247\char224\char241\char242\char232\char246\char224}
\textcyr{\char241\char236\char229\char249\char224\char229\char242\char241\char255}
\textcyr{\char237\char224} \textcyr{\char229\char228\char232\char237\char232\char246\char243}
\textcyr{\char226\char239\char240\char224\char226\char238} \textcyr{\char232\char235\char232}
\textcyr{\char226\char235\char229\char226\char238} \textcyr{\char241}
\textcyr{\char238\char228\char232\char237\char224\char234\char238\char226\char238\char233}
\textcyr{\char226\char229\char240\char238\char255\char242\char237\char238\char241\char242\char252\char254},
\textcyr{\char237\char229\char231\char224\char226\char232\char241\char232\char236\char238}
\textcyr{\char238\char242} \textcyr{\char239\char240\char229\char228\char251\char228\char243\char249\char232\char245}
\textcyr{\char239\char229\char240\char229\char236\char229\char249\char229\char237\char232\char233}.
\textcyr{\char207\char243\char241\char242\char252} $\xi(n)$ - \textcyr{\char234\char238\char238\char240\char228\char232\char237\char224\char242\char224}
\textcyr{\char247\char224\char241\char242\char232\char246\char251}
\textcyr{\char226} \textcyr{\char236\char238\char236\char229\char237\char242}
\textcyr{\char226\char240\char229\char236\char229\char237\char232}
$n=0,1,2,\,...$. \textcyr{\char205\char224\char233\char242\char232}
\textcyr{\char234\char238\char240\char240\char229\char235\char255\char246\char232\char238\char237\char237\char243\char254}
\textcyr{\char244\char243\char237\char234\char246\char232\char254}
\textcyr{\char241\char235\char243\char247\char224\char233\char237\char238\char227\char238}
\textcyr{\char239\char240\char238\char246\char229\char241\char241\char224}
$\xi(n)$.
\textbf{49.} \textcyr{\char209\char242\char224\char246\char232\char238\char237\char224\char240\char237\char251\char233}
\textcyr{\char226} \textcyr{\char248\char232\char240\char238\char234\char238\char236}
\textcyr{\char241\char236\char251\char241\char235\char229} \textcyr{\char241\char235\char243\char247\char224\char233\char237\char251\char233}
\textcyr{\char239\char240\char238\char246\char229\char241\char241}
$\xi(t)$ \textcyr{\char232\char236\char229\char229\char242} \textcyr{\char234\char238\char240\char240\char229\char235\char255\char246\char232\char238\char237\char237\char243\char254}
\textcyr{\char244\char243\char237\char234\char246\char232\char254}
$R_{\xi}(\tau)=e^{-\alpha|\tau|}(1-\beta|\tau|)$. \textcyr{\char205\char224\char233\char242\char232}
\textcyr{\char241\char239\char229\char234\char242\char240\char224\char235\char252\char237\char243\char254}
\textcyr{\char239\char235\char238\char242\char237\char238\char241\char242\char252}
\textcyr{\char236\char238\char249\char237\char238\char241\char242\char232}
\textcyr{\char253\char242\char238\char227\char238} \textcyr{\char241\char235\char243\char247\char224\char233\char237\char238\char227\char238}
\textcyr{\char239\char240\char238\char246\char229\char241\char241\char224}.
\textcyr{\char207\char240\char232} \textcyr{\char234\char224\char234\char232\char245}
\textcyr{\char231\char237\char224\char247\char229\char237\char232\char255\char245}
\textcyr{\char239\char224\char240\char224\char236\char229\char242\char240\char238\char226}
$(\alpha,\beta)$ \textcyr{\char234\char238\char240\char240\char229\char235\char255\char246\char232\char238\char237\char237\char224\char255}
\textcyr{\char244\char243\char237\char234\char246\char232\char255}
$R_{\xi}(\tau)$ \textcyr{\char232\char236\char229\char229\char242}
\textcyr{\char241\char236\char251\char241\char235}?$](https://dxdy-01.korotkov.co.uk/f/0/9/8/098344253956017408173963a9b98ea882.png "$% Предварительный просмотр исходного кода абзацев 0-2
\textbf{5.} \textcyr{\char205\char224\char233\char242\char232} \textcyr{\char238\char228\char237\char238\char236\char229\char240\char237\char243\char254}
\textcyr{\char239\char235\char238\char242\char237\char238\char241\char242\char252}
\textcyr{\char226\char229\char240\char238\char255\char242\char237\char238\char241\char242\char232}
\textcyr{\char241\char235\char243\char247\char224\char233\char237\char238\char227\char238}
\textcyr{\char239\char240\char238\char246\char229\char241\char241\char224}
$\xi(t)=\sin(\omega t+\varphi)$, \textcyr{\char229\char241\char235\char232}
\textcyr{\char241\char235\char243\char247\char224\char233\char237\char224\char255}
\textcyr{\char226\char229\char235\char232\char247\char232\char237\char224}
$\varphi$ \textcyr{\char240\char224\char241\char239\char240\char229\char228\char229\char235\char229\char237\char224}
\textcyr{\char240\char224\char226\char237\char238\char236\char229\char240\char237\char238}
\textcyr{\char237\char224} \textcyr{\char238\char242\char240\char229\char231\char234\char229}
$[-\pi,\pi]$, \textcyr{\char224} $\omega$ - \textcyr{\char228\char229\char242\char229\char240\char236\char232\char237\char232\char240\char238\char226\char224\char237\char237\char224\char255}
\textcyr{\char226\char229\char235\char232\char247\char232\char237\char224}.
\textbf{6.} \textcyr{\char194} \textcyr{\char237\char224\char247\char224\char235\char252\char237\char251\char233}
\textcyr{\char236\char238\char236\char229\char237\char242} \textcyr{\char226\char240\char229\char236\char229\char237\char232}
\textcyr{\char247\char224\char241\char242\char232\char246\char224}
\textcyr{\char237\char224\char245\char238\char228\char232\char242\char241\char255}
\textcyr{\char226} \textcyr{\char237\char224\char247\char224\char235\char229}
\textcyr{\char234\char238\char238\char240\char228\char232\char237\char224\char242}.
\textcyr{\char194} \textcyr{\char234\char224\char230\char228\char251\char233}
\textcyr{\char246\char229\char235\char238\char247\char232\char241\char235\char229\char237\char237\char251\char233}
\textcyr{\char236\char238\char236\char229\char237\char242} \textcyr{\char226\char240\char229\char236\char229\char237\char232}
\textcyr{\char247\char224\char241\char242\char232\char246\char224}
\textcyr{\char241\char236\char229\char249\char224\char229\char242\char241\char255}
\textcyr{\char237\char224} \textcyr{\char229\char228\char232\char237\char232\char246\char243}
\textcyr{\char226\char239\char240\char224\char226\char238} \textcyr{\char232\char235\char232}
\textcyr{\char226\char235\char229\char226\char238} \textcyr{\char241}
\textcyr{\char238\char228\char232\char237\char224\char234\char238\char226\char238\char233}
\textcyr{\char226\char229\char240\char238\char255\char242\char237\char238\char241\char242\char252\char254},
\textcyr{\char237\char229\char231\char224\char226\char232\char241\char232\char236\char238}
\textcyr{\char238\char242} \textcyr{\char239\char240\char229\char228\char251\char228\char243\char249\char232\char245}
\textcyr{\char239\char229\char240\char229\char236\char229\char249\char229\char237\char232\char233}.
\textcyr{\char207\char243\char241\char242\char252} $\xi(n)$ - \textcyr{\char234\char238\char238\char240\char228\char232\char237\char224\char242\char224}
\textcyr{\char247\char224\char241\char242\char232\char246\char251}
\textcyr{\char226} \textcyr{\char236\char238\char236\char229\char237\char242}
\textcyr{\char226\char240\char229\char236\char229\char237\char232}
$n=0,1,2,\,...$. \textcyr{\char205\char224\char233\char242\char232}
\textcyr{\char234\char238\char240\char240\char229\char235\char255\char246\char232\char238\char237\char237\char243\char254}
\textcyr{\char244\char243\char237\char234\char246\char232\char254}
\textcyr{\char241\char235\char243\char247\char224\char233\char237\char238\char227\char238}
\textcyr{\char239\char240\char238\char246\char229\char241\char241\char224}
$\xi(n)$.
\textbf{49.} \textcyr{\char209\char242\char224\char246\char232\char238\char237\char224\char240\char237\char251\char233}
\textcyr{\char226} \textcyr{\char248\char232\char240\char238\char234\char238\char236}
\textcyr{\char241\char236\char251\char241\char235\char229} \textcyr{\char241\char235\char243\char247\char224\char233\char237\char251\char233}
\textcyr{\char239\char240\char238\char246\char229\char241\char241}
$\xi(t)$ \textcyr{\char232\char236\char229\char229\char242} \textcyr{\char234\char238\char240\char240\char229\char235\char255\char246\char232\char238\char237\char237\char243\char254}
\textcyr{\char244\char243\char237\char234\char246\char232\char254}
$R_{\xi}(\tau)=e^{-\alpha|\tau|}(1-\beta|\tau|)$. \textcyr{\char205\char224\char233\char242\char232}
\textcyr{\char241\char239\char229\char234\char242\char240\char224\char235\char252\char237\char243\char254}
\textcyr{\char239\char235\char238\char242\char237\char238\char241\char242\char252}
\textcyr{\char236\char238\char249\char237\char238\char241\char242\char232}
\textcyr{\char253\char242\char238\char227\char238} \textcyr{\char241\char235\char243\char247\char224\char233\char237\char238\char227\char238}
\textcyr{\char239\char240\char238\char246\char229\char241\char241\char224}.
\textcyr{\char207\char240\char232} \textcyr{\char234\char224\char234\char232\char245}
\textcyr{\char231\char237\char224\char247\char229\char237\char232\char255\char245}
\textcyr{\char239\char224\char240\char224\char236\char229\char242\char240\char238\char226}
$(\alpha,\beta)$ \textcyr{\char234\char238\char240\char240\char229\char235\char255\char246\char232\char238\char237\char237\char224\char255}
\textcyr{\char244\char243\char237\char234\char246\char232\char255}
$R_{\xi}(\tau)$ \textcyr{\char232\char236\char229\char229\char242}
\textcyr{\char241\char236\char251\char241\char235}?$")
и постоянную величину добавки, что плотность будет такой же, как если под синусом стоит просто равномерно распределенная величина на ![$[-\pi,\pi]$](https://dxdy-04.korotkov.co.uk/f/7/3/0/73064caa415c5b3b6750a636bdb7978182.png "$[-\pi,\pi]$")
. Так что в результате просто вероятностная задача на преобразование случайных величин.