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zivnvUzFM=1g@!Iq!}Bk`s~`(!1mYsB_(|GM&r4!j;OZsaOa#hAI9N8IK9vSWTqzpP z`d(G*k%4y756p3k&v+*4f%2A;I@Bil-C;B;t49#Q@r6IpWq6!EQYKiG)*uj(9Ayx7 zC#Q++D*z&5?1nG>KoV-9#Sta*s&l!`?s8ywF;JETm4q3`TZ^HY^pBn2$uNYB!N}$B)nH*lHi7am3JGqT0 za!4|e+`g7=7+2ZkFGw>vMU`egw59<)!}S{=q;SZ^DeL%PhpKtT(KVwaks&bLoHX-U zR#nR&Q_53h{`QC?nul?EWvp9P1&F4$dhf4Mn(+uVrB2CCA0k`Q%BA*zxs*RJ{Ti>~ zQsdN(L}mR_E(5u$j$QU*S&=DxxRXY2`EmL|Ox8fetGMjHMRfS6i(@Nolpdy^kz+zi zwPI0$`GO0#GFJv*eUN$1>UL~j0nr_|q89DrsD7O9uCb5yIs&soS#!TMKL=iD@P{2+ zX;M`76A4Nih$S$6eUwO`iCdM~4>WARGkR%F0*$hTDiz*_XqJX?`h^k0)z6|D40?s1 zDOm#JUG5U{RMTk$YCO`u!GyROZbB}XBN}V!_(avBPO2I&Az$8C@6sX?r(!JYrzcMH zpNpfcS5FU(9z16~(#TA@vc}#5IG79S?1OOXCIf2@nw#&Ib-(NF=L-Ri_pSaCl1$Z* z+M+CVe}TDr8Zf@9r$p2FQ^)G6@y2J!IgQmzXx|rb=V|6iC&X=3rAix@Jn)5J!Wl;l z8o7>`$rHz&bn0;vJ5|2>YeO(DN(UB-L!u#rhNJ@r;!)92pQQtN%tp*o$_5t(^y>fr j4mq;dfB~H*O&&LK@|20k9pA~Qz$@w0yZ4ac`*iw0d!WCT diff --git a/probtheory/probtheory_2024_10_01.tex b/probtheory/probtheory_2024_10_01.tex index d30204e..300aaa0 100644 --- a/probtheory/probtheory_2024_10_01.tex +++ b/probtheory/probtheory_2024_10_01.tex @@ -190,8 +190,9 @@ Обозначим $\lambda_n = n p_n$. Тогда $p_n = \frac{\lambda_n}{n}$ и - $P_n(k) = C^k_n \left(\frac{\lambda_n}{n}\right)^k \left(1 - \frac{\lambda_n}{n}\right)^{n - k} \underset{n \to \infty}{\rightarrow} \frac{n^k}{k!} \frac{\lambda^k_n}{n^k} \left(1 - \frac{\lambda_n}{n}\right)^n \cancelto{1}{\left(1 - \frac{\lambda_n}{n}\right)^{-k}} - \underset{n \to \infty}{\rightarrow} \frac{\lambda_n^k}{k!} \left(1 - \frac{\lambda_n}{n}\right) = \frac{\lambda_n^k}{k!} \left(\left(1 - \frac{\lambda_n}{n}\right)^{-\frac{n}{\lambda_n}}\right)^{-\frac{\lambda_n}{n}n} \underset{n \to \infty}{\rightarrow} \frac{\lambda_n^k}{k!} e^{-\lambda_n} \underset{n \to \infty}{\rightarrow} \frac{\lambda^k}{k!} e^{-\lambda}$ + $P_n(k) = C^k_n \left(\frac{\lambda_n}{n}\right)^k \left(1 - \frac{\lambda_n}{n}\right)^{n - k} \underset{n \to \infty}{\rightarrow} + \frac{n^k}{k!} \frac{\lambda^k_n}{n^k} \left(1 - \frac{\lambda_n}{n}\right)^n \cancelto{1}{\left(1 - \frac{\lambda_n}{n}\right)^{-k}} + \underset{n \to \infty}{\rightarrow} \frac{\lambda_n^k}{k!} \left(1 - \frac{\lambda_n}{n}\right)^n = \frac{\lambda_n^k}{k!} \left(\left(1 - \frac{\lambda_n}{n}\right)^{-\frac{n}{\lambda_n}}\right)^{-\frac{\lambda_n}{n}n} \underset{n \to \infty}{\rightarrow} \frac{\lambda_n^k}{k!} e^{-\lambda_n} \underset{n \to \infty}{\rightarrow} \frac{\lambda^k}{k!} e^{-\lambda}$ $\Box$ \end{MyProof} diff --git a/probtheory/probtheory_2024_11_05.tex b/probtheory/probtheory_2024_11_05.tex index ef40112..c1012a3 100644 --- a/probtheory/probtheory_2024_11_05.tex +++ b/probtheory/probtheory_2024_11_05.tex @@ -54,11 +54,11 @@ \begin{MyProof} Пусть $a > 0$, тогда $F_\eta(x) = p(\eta < x) = p(a\xi + b < x) = p(\xi < \frac{x - b}{a}) = \int_{-\infty}^{\frac{x - b}{a}} f_\xi(t) dt = \left[\begin{matrix}t = \frac{y - b}{a} & dt = \frac{1}{a} dy & y = at + b \\ y(-\infty) = -\infty & y(\frac{x - b}{a}) = x\end{matrix}\right] = - \int_{-\infty}^x \frac{1}{a} f_\xi(\frac{y - b}{a}) dy \Longrightarrow \eta = \frac{1}{|a|} f_\xi (\frac{x - b}{a})$ + \int_{-\infty}^x \frac{1}{a} f_\xi(\frac{y - b}{a}) dy \Longrightarrow f_\eta(x) = \frac{1}{|a|} f_\xi (\frac{x - b}{a})$ Пусть $a < 0$, тогда $F_\eta(x) = p(\eta < x) = p(a\xi + b < x) = p(\xi > \frac{x - b}{a}) = \int_{\frac{x - b}{a}}^{\infty} f_\xi(t) dt = \left[\begin{matrix}t = \frac{y - b}{a} & dt = \frac{1}{a} dy & y = at + b \\ y(\infty) = -\infty & y(\frac{x - b}{a}) = x\end{matrix}\right] = - -\int_{-\infty}^x \frac{1}{a} f_\xi(\frac{y - b}{a}) dy \Longrightarrow \eta = \frac{1}{|a|} f_\xi (\frac{x - b}{a})$ + -\int_{-\infty}^x \frac{1}{a} f_\xi(\frac{y - b}{a}) dy \Longrightarrow f_\eta(x) = \frac{1}{|a|} f_\xi (\frac{x - b}{a})$ \end{MyProof} \underline{Следствие} @@ -170,7 +170,7 @@ 4) Если существует момент $m_t$ случайной величины $\xi$, то существует $m_s$ при $s < t$ (при условии, что интеграл/сумма сходятся) \begin{MyProof} - Пусть $s < t$. Тогда $|x|^s \leq \min(1, |x|^t) \leq 1 + |x|^t$, так как при $|x| < 1, \ |x|^s \leq 1$ и при $|x| \geq 1, \ |x|^s \leq |x|^t$ + Пусть $s < t$. Тогда $|x|^s \leq \max(1, |x|^t) \leq 1 + |x|^t$, так как при $|x| < 1, \ |x|^s \leq 1$ и при $|x| \geq 1, \ |x|^s \leq |x|^t$ $E|\xi|^s \leq E|\xi|^t + 1$ и если $E|\xi|^t$ существует (конечно), то $\exists E|\xi|^s$ diff --git a/probtheory/probtheory_2024_11_12.tex b/probtheory/probtheory_2024_11_12.tex index 6cebc46..147fb78 100644 --- a/probtheory/probtheory_2024_11_12.tex +++ b/probtheory/probtheory_2024_11_12.tex @@ -24,7 +24,7 @@ \Nota То есть $p(\xi \text{ не имеет свойство } \mathrm{Cond}) = 0$ - $p(\omega \in \Omega \ | \ \xi(\omega) \text{ не имеет св-во } \mathrm{Cond})$ + $p(\omega \in \Omega \ | \ \xi(\omega) \text{ не имеет св-во } \mathrm{Cond}) = 0$ \Def Последовательность случайных величин $\{\xi_n\}$ сходится \enquote{почти наверное} к случайной величине $\xi$ при $n \to \infty$ ($\xi_n \overset{\text{п. н.}}{\longrightarrow} \xi$), если $p(\omega \in \Omega \ | \ \xi_n(\omega) \underset{n \to \infty}{\longrightarrow} \xi(\omega)) = 1$ @@ -98,7 +98,7 @@ $E|\xi| \geq E(\varepsilon \cdot I(|\xi| \geq \varepsilon))$ - $E|\xi| \geq \varepsilon \cdot E(\varepsilon I(|\xi| \geq \varepsilon)) = \varepsilon \cdot p(|\xi| \geq \varepsilon) + $E|\xi| \geq \varepsilon \cdot E(I(|\xi| \geq \varepsilon)) = \varepsilon \cdot p(|\xi| \geq \varepsilon) \Longrightarrow p(|\xi| \geq \varepsilon) \leq \frac{E|\xi|}{\varepsilon}$ \end{MyProof} @@ -111,7 +111,7 @@ \end{MyTheorem} \begin{MyProof} - $p(|\xi - E\xi| \geq \varepsilon) = p((\xi - E\xi)^2 \geq \varepsilon^2) \leq \frac{E(\xi - E\xi)^2}{\varepsilon^2} = \frac{D\xi}{\varepsilon}$ + $p(|\xi - E\xi| \geq \varepsilon) = p((\xi - E\xi)^2 \geq \varepsilon^2) \leq \frac{E(\xi - E\xi)^2}{\varepsilon^2} = \frac{D\xi}{\varepsilon^2}$ \end{MyProof} \subsubsection{III. Правило \enquote{трех сигм}} diff --git a/probtheory/probtheory_2024_11_19.tex b/probtheory/probtheory_2024_11_19.tex index ba00ade..fa72f3b 100644 --- a/probtheory/probtheory_2024_11_19.tex +++ b/probtheory/probtheory_2024_11_19.tex @@ -183,7 +183,7 @@ \begin{MyProof} $F_{\xi}(x) = \lim_{y \to \infty} F_{\xi, \eta}(x, y) = \int_{-\infty}^x \int_{-\infty}^\infty f(x, y) dydx$ - Из этого $\int_{-\infty}^\infty f(x, y) = f_\xi(x)$ + Из этого $\int_{-\infty}^\infty f(x, y) dy = f_\xi(x)$ \end{MyProof} \item Так как вероятность попадания в Борелевские множества полностью задается функцией распределения, diff --git a/probtheory/probtheory_2024_12_03.tex b/probtheory/probtheory_2024_12_03.tex index 24d62b8..6b0cb1e 100644 --- a/probtheory/probtheory_2024_12_03.tex +++ b/probtheory/probtheory_2024_12_03.tex @@ -69,7 +69,7 @@ \subsection{Пространство случайных величин} \begin{MyProof} $P_2(x) = E(x\xi - \eta)^2 = x^2 E\xi^2 - 2xE(\xi\eta) + E\eta^2 \geq 0 \Longrightarrow D = 4(E(\xi\eta))^2 - - 4 E\xi^2 - E\eta^2 \leq 0 \Longrightarrow |E(\xi\eta)| \leq \sqrt{E\xi^2 \cdot E\eta^2}$ + 4 E\xi^2 \cdot E\eta^2 \leq 0 \Longrightarrow |E(\xi\eta)| \leq \sqrt{E\xi^2 \cdot E\eta^2}$ $|E(\xi, \eta)| = \sqrt{E\xi^2 - E\eta^2} \Longrightarrow D = 0 \Longrightarrow \exists$ какая-либо точка касания $C$, из этого $E(C\xi - \eta)^2 = 0 \Longrightarrow C\xi - \eta = 0 \Longleftrightarrow \eta = C\xi \text{ п.н. }$ @@ -191,7 +191,7 @@ \subsection{Коэффициент линейной корреляции} \hypertarget{correlation}{} \Def Коэффициентом корреляции случайных величин $\xi$ и $\eta$ с конечными вторыми моментами, -называется величина $r_{\xi,\eta} = \frac{\mathrm{cov(\xi, \eta)}}{\sqrt{D\xi} \sqrt{D\eta}} = \frac{E(\xi\eta) - E\xiE\eta}{\sigma_\xi \sigma_\eta}$ +называется величина $r_{\xi,\eta} = \frac{\mathrm{cov(\xi, \eta)}}{\sqrt{D\xi} \sqrt{D\eta}} = \frac{E(\xi\eta) - E\xi E\eta}{\sigma_\xi \sigma_\eta}$ Можно записать в другой форме: $r_{\xi,\eta} = \frac{E((\xi - E\xi)(\eta - E\eta))}{\sqrt{E(\xi - E\xi)^2}\sqrt{E(\eta - E\eta)^2}} = \frac{(\xi - E\xi, \eta - E\eta)}{\|\xi - E\xi\|\|\eta - E\eta\|} = \cos(\widehat{\xi - E\xi, \eta - E\eta})$ - косинус угла между величинами (грубая интерпретация) diff --git a/probtheory/probtheory_2024_12_10.tex b/probtheory/probtheory_2024_12_10.tex index 2a401e2..ab4c985 100644 --- a/probtheory/probtheory_2024_12_10.tex +++ b/probtheory/probtheory_2024_12_10.tex @@ -58,7 +58,7 @@ \subsection{Характеристические функции} \begin{MyProof} $\varphi_\xi(t) = Ee^{it\xi} = E(1 + it\xi + \frac{(it\xi)^2}{2!} + \dots + \frac{(it\xi)^k}{k!} + o(|t|^k)) = - 1 + it E\xi \frac{i^2 t^2}{2}E\xi^2 + \dots + \frac{(it)^k}{k!} E\xi^k + o(|t|^k)$ + 1 + it E\xi + \frac{i^2 t^2}{2}E\xi^2 + \dots + \frac{(it)^k}{k!} E\xi^k + o(|t|^k)$ \end{MyProof} \item Пусть $E\xi^k < \infty$. Тогда $\varphi_\xi^{(k)}(0) = i^k E\xi^k$ @@ -217,7 +217,7 @@ \subsubsection{Центральная предельная теорема} Обозначим $Z_n = \eta_1 + \dots + \eta_n = \frac{(\xi_1 + \dots + \xi_n) - na}{\sigma} = \frac{S_n - na}{\sigma}$ - Надо доказать, что если $\frac{Z_n}{\sqrt{n}} \rightrightarrows N(0, 1)$ + Надо доказать, что $\frac{Z_n}{\sqrt{n}} \rightrightarrows N(0, 1)$ По четвертому свойству $\varphi_{\eta_1}(t) = 1 + itE\eta_1 - \frac{t^2}{2} E\eta_1^2 + o(t^2) = 1 - \frac{t^2}{2} + o(t^2)$ diff --git a/probtheory/probtheory_superconspect.tex b/probtheory/probtheory_superconspect.tex index e88c676..46b2c96 100644 --- a/probtheory/probtheory_superconspect.tex +++ b/probtheory/probtheory_superconspect.tex @@ -1011,8 +1011,9 @@ Обозначим $\lambda_n = n p_n$. Тогда $p_n = \frac{\lambda_n}{n}$ и - $P_n(k) = C^k_n \left(\frac{\lambda_n}{n}\right)^k \left(1 - \frac{\lambda_n}{n}\right)^{n - k} \underset{n \to \infty}{\rightarrow} \frac{n^k}{k!} \frac{\lambda^k_n}{n^k} \left(1 - \frac{\lambda_n}{n}\right)^n \cancelto{1}{\left(1 - \frac{\lambda_n}{n}\right)^{-k}} - \underset{n \to \infty}{\rightarrow} \frac{\lambda_n^k}{k!} \left(1 - \frac{\lambda_n}{n}\right) = \frac{\lambda_n^k}{k!} \left(\left(1 - \frac{\lambda_n}{n}\right)^{-\frac{n}{\lambda_n}}\right)^{-\frac{\lambda_n}{n}n} \underset{n \to \infty}{\rightarrow} \frac{\lambda_n^k}{k!} e^{-\lambda_n} \underset{n \to \infty}{\rightarrow} \frac{\lambda^k}{k!} e^{-\lambda}$ + $P_n(k) = C^k_n \left(\frac{\lambda_n}{n}\right)^k \left(1 - \frac{\lambda_n}{n}\right)^{n - k} \underset{n \to \infty}{\rightarrow} + \frac{n^k}{k!} \frac{\lambda^k_n}{n^k} \left(1 - \frac{\lambda_n}{n}\right)^n \cancelto{1}{\left(1 - \frac{\lambda_n}{n}\right)^{-k}} + \underset{n \to \infty}{\rightarrow} \frac{\lambda_n^k}{k!} \left(1 - \frac{\lambda_n}{n}\right)^n = \frac{\lambda_n^k}{k!} \left(\left(1 - \frac{\lambda_n}{n}\right)^{-\frac{n}{\lambda_n}}\right)^{-\frac{\lambda_n}{n}n} \underset{n \to \infty}{\rightarrow} \frac{\lambda_n^k}{k!} e^{-\lambda_n} \underset{n \to \infty}{\rightarrow} \frac{\lambda^k}{k!} e^{-\lambda}$ $\Box$ \end{MyProof} @@ -2146,11 +2147,11 @@ \begin{MyProof} Пусть $a > 0$, тогда $F_\eta(x) = p(\eta < x) = p(a\xi + b < x) = p(\xi < \frac{x - b}{a}) = \int_{-\infty}^{\frac{x - b}{a}} f_\xi(t) dt = \left[\begin{matrix}t = \frac{y - b}{a} & dt = \frac{1}{a} dy & y = at + b \\ y(-\infty) = -\infty & y(\frac{x - b}{a}) = x\end{matrix}\right] = - \int_{-\infty}^x \frac{1}{a} f_\xi(\frac{y - b}{a}) dy \Longrightarrow \eta = \frac{1}{|a|} f_\xi (\frac{x - b}{a})$ + \int_{-\infty}^x \frac{1}{a} f_\xi(\frac{y - b}{a}) dy \Longrightarrow f_\eta(x) = \frac{1}{|a|} f_\xi (\frac{x - b}{a})$ Пусть $a < 0$, тогда $F_\eta(x) = p(\eta < x) = p(a\xi + b < x) = p(\xi > \frac{x - b}{a}) = \int_{\frac{x - b}{a}}^{\infty} f_\xi(t) dt = \left[\begin{matrix}t = \frac{y - b}{a} & dt = \frac{1}{a} dy & y = at + b \\ y(\infty) = -\infty & y(\frac{x - b}{a}) = x\end{matrix}\right] = - -\int_{-\infty}^x \frac{1}{a} f_\xi(\frac{y - b}{a}) dy \Longrightarrow \eta = \frac{1}{|a|} f_\xi (\frac{x - b}{a})$ + -\int_{-\infty}^x \frac{1}{a} f_\xi(\frac{y - b}{a}) dy \Longrightarrow f_\eta(x) = \frac{1}{|a|} f_\xi (\frac{x - b}{a})$ \end{MyProof} \underline{Следствие} @@ -2262,7 +2263,7 @@ 4) Если существует момент $m_t$ случайной величины $\xi$, то существует $m_s$ при $s < t$ (при условии, что интеграл/сумма сходятся) \begin{MyProof} - Пусть $s < t$. Тогда $|x|^s \leq \min(1, |x|^t) \leq 1 + |x|^t$, так как при $|x| < 1, \ |x|^s \leq 1$ и при $|x| \geq 1, \ |x|^s \leq |x|^t$ + Пусть $s < t$. Тогда $|x|^s \leq \max(1, |x|^t) \leq 1 + |x|^t$, так как при $|x| < 1, \ |x|^s \leq 1$ и при $|x| \geq 1, \ |x|^s \leq |x|^t$ $E|\xi|^s \leq E|\xi|^t + 1$ и если $E|\xi|^t$ существует (конечно), то $\exists E|\xi|^s$ @@ -2341,7 +2342,7 @@ \Nota То есть $p(\xi \text{ не имеет свойство } \mathrm{Cond}) = 0$ - $p(\omega \in \Omega \ | \ \xi(\omega) \text{ не имеет св-во } \mathrm{Cond})$ + $p(\omega \in \Omega \ | \ \xi(\omega) \text{ не имеет св-во } \mathrm{Cond}) = 0$ \Def Последовательность случайных величин $\{\xi_n\}$ сходится \enquote{почти наверное} к случайной величине $\xi$ при $n \to \infty$ ($\xi_n \overset{\text{п. н.}}{\longrightarrow} \xi$), если $p(\omega \in \Omega \ | \ \xi_n(\omega) \underset{n \to \infty}{\longrightarrow} \xi(\omega)) = 1$ @@ -2415,7 +2416,7 @@ $E|\xi| \geq E(\varepsilon \cdot I(|\xi| \geq \varepsilon))$ - $E|\xi| \geq \varepsilon \cdot E(\varepsilon I(|\xi| \geq \varepsilon)) = \varepsilon \cdot p(|\xi| \geq \varepsilon) + $E|\xi| \geq \varepsilon \cdot E(I(|\xi| \geq \varepsilon)) = \varepsilon \cdot p(|\xi| \geq \varepsilon) \Longrightarrow p(|\xi| \geq \varepsilon) \leq \frac{E|\xi|}{\varepsilon}$ \end{MyProof} @@ -2428,7 +2429,7 @@ \end{MyTheorem} \begin{MyProof} - $p(|\xi - E\xi| \geq \varepsilon) = p((\xi - E\xi)^2 \geq \varepsilon^2) \leq \frac{E(\xi - E\xi)^2}{\varepsilon^2} = \frac{D\xi}{\varepsilon}$ + $p(|\xi - E\xi| \geq \varepsilon) = p((\xi - E\xi)^2 \geq \varepsilon^2) \leq \frac{E(\xi - E\xi)^2}{\varepsilon^2} = \frac{D\xi}{\varepsilon^2}$ \end{MyProof} \subsubsection{III. Правило \enquote{трех сигм}} @@ -2734,7 +2735,7 @@ \begin{MyProof} $F_{\xi}(x) = \lim_{y \to \infty} F_{\xi, \eta}(x, y) = \int_{-\infty}^x \int_{-\infty}^\infty f(x, y) dydx$ - Из этого $\int_{-\infty}^\infty f(x, y) = f_\xi(x)$ + Из этого $\int_{-\infty}^\infty f(x, y) dy = f_\xi(x)$ \end{MyProof} \item Так как вероятность попадания в Борелевские множества полностью задается функцией распределения, @@ -3061,7 +3062,7 @@ \subsection{Пространство случайных величин} \begin{MyProof} $P_2(x) = E(x\xi - \eta)^2 = x^2 E\xi^2 - 2xE(\xi\eta) + E\eta^2 \geq 0 \Longrightarrow D = 4(E(\xi\eta))^2 - - 4 E\xi^2 - E\eta^2 \leq 0 \Longrightarrow |E(\xi\eta)| \leq \sqrt{E\xi^2 \cdot E\eta^2}$ + 4 E\xi^2 \cdot E\eta^2 \leq 0 \Longrightarrow |E(\xi\eta)| \leq \sqrt{E\xi^2 \cdot E\eta^2}$ $|E(\xi, \eta)| = \sqrt{E\xi^2 - E\eta^2} \Longrightarrow D = 0 \Longrightarrow \exists$ какая-либо точка касания $C$, из этого $E(C\xi - \eta)^2 = 0 \Longrightarrow C\xi - \eta = 0 \Longleftrightarrow \eta = C\xi \text{ п.н. }$ @@ -3183,7 +3184,7 @@ \subsection{Коэффициент линейной корреляции} \hypertarget{correlation}{} \Def Коэффициентом корреляции случайных величин $\xi$ и $\eta$ с конечными вторыми моментами, -называется величина $r_{\xi,\eta} = \frac{\mathrm{cov(\xi, \eta)}}{\sqrt{D\xi} \sqrt{D\eta}} = \frac{E(\xi\eta) - E\xiE\eta}{\sigma_\xi \sigma_\eta}$ +называется величина $r_{\xi,\eta} = \frac{\mathrm{cov(\xi, \eta)}}{\sqrt{D\xi} \sqrt{D\eta}} = \frac{E(\xi\eta) - E\xi E\eta}{\sigma_\xi \sigma_\eta}$ Можно записать в другой форме: $r_{\xi,\eta} = \frac{E((\xi - E\xi)(\eta - E\eta))}{\sqrt{E(\xi - E\xi)^2}\sqrt{E(\eta - E\eta)^2}} = \frac{(\xi - E\xi, \eta - E\eta)}{\|\xi - E\xi\|\|\eta - E\eta\|} = \cos(\widehat{\xi - E\xi, \eta - E\eta})$ - косинус угла между величинами (грубая интерпретация) @@ -3298,7 +3299,7 @@ \subsection{Характеристические функции} \begin{MyProof} $\varphi_\xi(t) = Ee^{it\xi} = E(1 + it\xi + \frac{(it\xi)^2}{2!} + \dots + \frac{(it\xi)^k}{k!} + o(|t|^k)) = - 1 + it E\xi \frac{i^2 t^2}{2}E\xi^2 + \dots + \frac{(it)^k}{k!} E\xi^k + o(|t|^k)$ + 1 + it E\xi + \frac{i^2 t^2}{2}E\xi^2 + \dots + \frac{(it)^k}{k!} E\xi^k + o(|t|^k)$ \end{MyProof} \item Пусть $E\xi^k < \infty$. Тогда $\varphi_\xi^{(k)}(0) = i^k E\xi^k$ @@ -3457,7 +3458,7 @@ \subsubsection{Центральная предельная теорема} Обозначим $Z_n = \eta_1 + \dots + \eta_n = \frac{(\xi_1 + \dots + \xi_n) - na}{\sigma} = \frac{S_n - na}{\sigma}$ - Надо доказать, что если $\frac{Z_n}{\sqrt{n}} \rightrightarrows N(0, 1)$ + Надо доказать, что $\frac{Z_n}{\sqrt{n}} \rightrightarrows N(0, 1)$ По четвертому свойству $\varphi_{\eta_1}(t) = 1 + itE\eta_1 - \frac{t^2}{2} E\eta_1^2 + o(t^2) = 1 - \frac{t^2}{2} + o(t^2)$