From 4b8c9085a135deb9440e144e0dae885737839bea Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Mon, 12 Jul 2021 23:04:27 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=E9=9A=8F=E6=9C=BA=E5=8F=98?= =?UTF-8?q?=E9=87=8F=E4=B8=8E=E5=88=86=E5=B8=83?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- model/model.pdf | Bin 12380 -> 0 bytes .../random-variables-and-distribution.pdf | Bin 12380 -> 53219 bytes .../random-variables-and-distribution.tex | 74 ++++++ .../3-digital-features/digital-features.pdf | Bin 12380 -> 0 bytes .../3-digital-features/digital-features.tex | 38 +++ ...arge-numbers-and-central-limit-theorem.pdf | Bin 12380 -> 0 bytes ...arge-numbers-and-central-limit-theorem.tex | 38 +++ .../mathematical-statistics.pdf | Bin 12380 -> 0 bytes .../mathematical-statistics.tex | 38 +++ .../random-events-and-probability.pdf | Bin 71822 -> 126669 bytes .../random-events-and-probability.tex | 166 +++++++++++- 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G(p)$。 + +又$p=P\{X\geqslant3\}=\int_3^{+\infty}2^{-x}\ln2\,\textrm{d}x=\dfrac{1}{8}$ + +这是对几何分布的变形,首先进行$k$次试验,第$k$次成功,所以要乘$p$,而因为是第2个成功,所以前面的$k-1$次中有$k-2$次失败和一次成功,所以一共$p^2(1-k)^{k-2}$。因为前面的成功的一次在$k-1$中任意一个地方就可以了,所以一共有$k-1$中可能性,要考虑到排列,所以还要乘$(k-1)$。 + +$\therefore P\{Y=k\}=(k-1)\left(\dfrac{1}{8}\right)^2\cdot\left(\dfrac{7}{8}\right)^{k-2}$。 + +\end{document} diff --git a/probability-theory-and-mathematical-statistics/exercise/3-digital-features/digital-features.pdf b/probability-theory-and-mathematical-statistics/exercise/3-digital-features/digital-features.pdf deleted file mode 100644 index a18f2644c1530b98343259cf5b65529efd2cd1a3..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 12380 zcmcI~WpF0RvYnWjMk8is=@Tt|>#LUdh%*^1u_H(>lzx93VAMVKL zsP3-FsydxfS$Q%^<%LCQ8R-Bpr2E$w&oGQE1PlarhL$kg-1MRr)=nl4^rF@VPA0-8 zMs~&~^wK7_W=`e=j2r-7UKmFw2NMGu7`N46IYt-|0l0H-5ng@iy2K40Bv=7(t{u+W z8LldtcVDARSpk7K(sApw?_~D`q~WE?QIfVTtWLj&48J*gPF(JNVK33rHc66Y)pIm1 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a/probability-theory-and-mathematical-statistics/knowledge/1-random-events-and-probability/random-events-and-probability.tex b/probability-theory-and-mathematical-statistics/knowledge/1-random-events-and-probability/random-events-and-probability.tex index 960f031..a5327e6 100644 --- a/probability-theory-and-mathematical-statistics/knowledge/1-random-events-and-probability/random-events-and-probability.tex +++ b/probability-theory-and-mathematical-statistics/knowledge/1-random-events-and-probability/random-events-and-probability.tex @@ -25,6 +25,8 @@ % 数学公式 \usepackage[colorlinks,linkcolor=black,urlcolor=blue]{hyperref} % 超链接 +\usepackage{pifont} +% 圆圈序号 \author{Didnelpsun} \title{随机事件与概率} \date{} @@ -116,18 +118,180 @@ \begin{itemize} \item 描述性定义:将随机事件$A$发生的可能性大小的度量(非负)称为事件$A$发生的概率,记为$P(A)$。 \item 统计性定义:在相同条件下做重复试验,事件$A$出现的次数$k$和总的试验次数$n$之比$\dfrac{k}{n}$,称为事件$A$在这$n$次试验中出现的\textbf{频率},当$n$充分大时,频率将稳定与某常数$p$附近,$n$越大频率偏离这个常数$p$的可能性越小,这个常数$p$就是事件$A$的概率。 + \item 公理化定义:设随机试验的样本空间为$\Omega$,如果对每一个事件$A$都有一个确定的实数$P(A)$,且事件函数$P(\cdot)$满足:\ding{172}非负性:$P(A)\geqslant0$;\ding{173}规范性:$P(\Omega)=1$;\ding{174}可列可加性:对于任意个互不相容事件$A_1,A_2,\cdots,A_n$有$P(\bigcup\limits_{i=1}^\infty A_i)=\sum\limits_{i=1}^nP(A_i)$,则称$P(\cdot)$为概率,$P(A)$为事件$A$的概率。 \end{itemize} -\subsection{概率模型} +\subsection{概率类型} \subsubsection{古典概型} +\textcolor{violet}{\textbf{定义:}}样本空间满足:\ding{172}只有有限个样本点(基本事件);\ding{173}每个样本点(基本事件)发生的可能性一样(等可能)。 + +若古典概型的基本事件总数为$n$,事件$A$包含$k$个基本事件,也称为有利于$A$的基本事件为$k$个,则$A$的概率为$P(A)=\dfrac{k}{n}=\dfrac{\text{事件}A\text{所含基本事件的个数}}{\text{基本事件总数}}$,这个概率就是$A$的\textbf{古典概率}。 + +古典概型的关键是计数,常用的方法有三种: + +\begin{enumerate} + \item 列举法(直接查数法):基本事件为数不多使用。 + \item 集合对应法:\begin{enumerate} + \item 加法原理:完成一件事有$n$类方法,第一类方法中有$m_1$类方法,第二类办法有$m_2$中方法,$\cdots$,第$n$类方法中有$m_n$类方法,所以完成此事共有$\sum\limits_{i=1}^nm_i$种方法。 + \item 乘法原理:完成一件事情有$n$个步骤,第一步有$m_1$种方法,第二步有$m_2$种方法,$\cdots$,第$n$步有$m_n$种方法,则完成此事共有$\prod\limits_{i=1}^nm_i$种方法。 + \item 排列:从$n$种不同的元素种取出$m\leqslant n$个元素,并按照一定顺序排成一列,称为排列,所有排列的个数称为排列数,记为$P_n^m=n(n-1)(n-2)\cdots(n-m+1)=\dfrac{n!}{(n-m)!}$,当$m=n$时,$P_m^n=n!$称为\textbf{全排列}。 + \item 组合:从$n$种不同的元素种取出$m\leqslant n$个元素,并组成一组,称为组合,所有组合的个数称为组合数,记为$C_n^m=\dfrac{P_n^m}{m!}=\dfrac{n!}{(n-m)!m!}$。 + \end{enumerate} + \item 逆数法:先求$\overline{A}$中的基本事件数$n_{\overline{A}}$,将基本事件总数$n$减去$n_{\overline{A}}$得$A$中的基本事件数。常用于计算含有“至少”字样的事件的概率。 +\end{enumerate} + +问题常见类型: + +\begin{itemize} + \item 直接用定义求概率。 + \item 随机分配或随机占位。将$n$个可辨质点是随机分配到$N$个盒子中。若每盒最多可容纳一个质点,则一共有$P_N^n$种分法;若每盒可以容纳任意多个质点,则一共有$N^n$种分法。 + \item 简单随机抽样。设$\Omega=\{\omega_1,\omega_2,\cdots,\omega_N\}$含有$N$个元素,称$\Omega$为总体。各元素被抽到的可能性相同。若先后有放回取$n$次,则有$N^n$种抽法;若先后无放回取$n$次,则有$P_N^n$种抽法;若任取$n$个,则有$C_N^n$种抽法。 +\end{itemize} + +\textbf{例题:}5人共钓到3条鱼,每条鱼每个人钓到的可能性相同,求: + +(1)3条鱼由不同人钓到的概率。 + +(2)有1人钓到两条鱼的概率。 + +(3)3条鱼由同一个人钓到的概率。 + +由题目可知这是一个随机分配的问题,总基本事件数为$5^3$。 + +对于鱼而言没有明确的区分说明,所以这个就是个组合问题。 + +(1)解:令第一个事件为$A$,因为每条鱼由不同的人钓到,即5人中恰有3人各钓到鱼,所以组合一共有$C_5^3$种,即从5个人取3个人有这么多种的取法。这3个人需要钓到3条鱼,因为鱼是可辩的,所以每组有$P_3^3$种分配方法。则$P(A)=\dfrac{C_5^3P_3^3}{5^3}$。 + +(2)解:令第二个事件为$B$,若一个人钓到两条,即从3条中任意选2,即$C_3^2$,又是5个人中的一个人完成的,所以$C_5^1$,所以有一个人钓到2条鱼共有$C_3^2C_5^1$种可能,此时还有一条鱼可以被其他4个人钓到,所以还要乘4。则$P(B)=\dfrac{C_3^2C_5^14}{5^3}$。 + +(3)解:令第三个事件为$C$,若一个人钓到三条,所以只有一种选法,然后有5个人可能钓到3条,所以是$C_5^1$,则$P(C)=\dfrac{C_5^1}{5^3}$。 + \subsubsection{几何概型} +\textcolor{violet}{\textbf{定义:}}\ding{172}样本空间$\Omega$是一个可度量的有界区域;\ding{173}每个样本点发生的可能性都是一样,即样本点落入$\Omega$的某一可度量的子区域$S$的可能性大小与$S$的几何度量成正比,而与$S$的位置与形状无关。 + +在几何概型随机试验中,若$S_A$是样本空间$\Omega$的一个可度量的子区域,则事件$A={\text{样本落入区域}S_A}$的概率为$P(A)=\dfrac{S_A\text{的几何度量}}{\Omega\text{的几何度量}}$,这个概率就是$A$的\textbf{几何概率}。 + +古典概型的基本事件有限,而几何概型的基本事件无限且可几何度量。 + \subsection{性质} +\begin{itemize} + \item 有界性:对于任一事件$A$,有$0\leqslant P(A)\leqslant1$,且$P(\varnothing)=0$,$P(\Omega)=1$。($P(A)=0$不能推出$A=\varnothing$,同样$P(A)=1$不能推出$A=\Omega$) + \item 单调性:设$AB$为两个事件,若$A\subset B$,则$P(B-A)=P(B)-P(A)$,$P(B)\geqslant P(A)$。 +\end{itemize} + \subsection{公式} +\begin{itemize} + \item 逆事件概率公式:对于任一事件$A$,有$P(\overline{A})=1-P(A)$。 + \item 加法公式:对于任意两个事件$AB$,有$P(A\cup B)=P(A)+P(B)-P(AB)$;对于三个事件$ABC$,$P(A\cup B\cup C)=P(A)+P(B)+P(C)-P(AB)-P(AC)-P(BC)+P(ABC)$;对于四个事件$ABCD$,$P(A\cup B\cup C\cup D)=P(A)+P(B)+P(C)+P(D)-P(AB)-P(AC)-P(AD)-P(BC)-P(BD)-P(CD)+P(ABC)+P(ABD)+P(ACD)+P(BCD)-P(ABCD)$;若$A_1,A_2,\cdots,A_n$两两互不相容,则$P(A_1\cup A_2\cup\cdots\cup A_n)=P(A_1)+P(A_2)+\cdots+P(A_n)$。 + \item 减法公式:$P(A-B)=P(A)-P(AB)=P(A\overline{B})$。 + \item 条件概率公式:对于任意两个事件$AB$,若$P(A)>0$,我们称在已知事件$A$发生的条件下,事件$B$发生的概率为\textbf{条件概率},记为$P(B|A)=\dfrac{P(AB)}{P(A)}$。$P(\overline{B}|A)=1-P(B|A)$,$P(B-C|A)=P(B|A)-P(BC|A)$。 + \item 乘法公式:若$P(A)>0$,则$P(AB)=P(A)P(B|A)$。一般而言,对于$n>2$,$P(A_1A_2\cdots A_{n-1})>0$,则$P(A_1A_2\cdots A_{n-1})=P(A_1)P(A_2|A_1)P(A_3|A_1A_2)$\\$\cdots P(A_n|A_1A_2\cdots A_{n-1})$。($A_i$的顺序不定) + \item 全概率公式:若$\bigcup\limits_{i=1}^nA_i=\Omega$,$A_iA_j=\varnothing$($i\neq j$,$i,j=1,2,\cdots,n$),$P(A_i)>0$,则对任一事件$B$,有$B=\bigcup\limits_{i=1}^nA_iB$,$P(B)=\sum\limits_{i=1}^nP(A_i)P(B|A_i)$。$P(B)=P(B\Omega)=P(B(\bigcup\limits_{i=1}^nA_i))=P(\bigcup\limits_{i=1}^nBA_i)=\sum\limits_{i=1}^nP(BA_i)=\sum\limits_{i=1}^nP(A_i)P(B|A_i)$。 + \item 贝叶斯公式:若$\bigcup\limits_{i=1}^nA_i=\Omega$,$A_iA_j=\varnothing$($i\neq j$,$i,j=1,2,\cdots,n$),$P(A_i)>0$,则对任一事件$B$,有$P(A_i|B)=\dfrac{P(A_iB)}{P(B)}=\dfrac{P(A_i)P(B|A_i)}{\sum\limits_{i=1}^nP(A_i)P(B|A_i)}$。 +\end{itemize} + +全概率公式是由因知果,而贝叶斯公式是执果索因。 + +\textbf{例题:}若随机事件$AB$同时发生,$C$也必然发生,且下列选项一定成立的是()。 + +$A.P(C)LHFwMIDAHlmPc6Hx%X=%gExd2v_pkSY z_cv55phIklM|$RQV?&6UQ7xgvnx8SU9arF&+7L9LPXmAU<~=R&0ZwD9TVG!9CD&?I z)>NFB>-tsG@^$(w{qqV808bG8#mD?p-2Mv|Z~y~vTH@|~cT5Z3f3j;Q;7 zknWbw9+TBmQpoJ4<<#OCX>+~_GR}!oD$=$<(l@VAhR(-CXHJG29)@UgvEYVC6_V37 z;S*oF0x*}{h}xpjF;QhjK>vSP;@fr#*}>It(7@$YaGD@tJs;{UuX|^az=p7G;cKEx z9tIf-eTvLLIdcU$;iDroJimkf9t06u^|F6E6>n2LP(b-wyHipA!slWO>K2rFB8K_+ 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+\textcolor{violet}{\textbf{定义:}}如果$X$的概率分布为$P\{X=k\}=(1-p)^{k-1}p$($k=0,1,\cdots,n$,$0t|>#LUdh%*^1u_H(>lzx93VAMVKL zsP3-FsydxfS$Q%^<%LCQ8R-Bpr2E$w&oGQE1PlarhL$kg-1MRr)=nl4^rF@VPA0-8 zMs~&~^wK7_W=`e=j2r-7UKmFw2NMGu7`N46IYt-|0l0H-5ng@iy2K40Bv=7(t{u+W z8LldtcVDARSpk7K(sApw?_~D`q~WE?QIfVTtWLj&48J*gPF(JNVK33rHc66Y)pIm1 zVJt8V-#^QJnccXqaX%9ktJ4zdpP!P6$W2U;>N)a~~`R^JBHpi1;82tmf< zaJ%;TjIT8)!IINjpQdRlhiDY39hgcsPUh)O*)1)>2eSlS8j)93LOB-Ig8q`V{mQNx zkGtP`9qc=YCa+YXf-$i*{*TrC(fZf&8QK034vc@n!8!6@IG_o6%EiN;yZX2cjdvRo zR>HdUUiC}PjP|~1F;Z5Au6-b`^a-A?_gXtFsQr;kXyZ6i`2I ztq2$rrA8bnWe&&>m9;prXGuhbqJ5N1q=768>MDGW@YhOEzFN~)S|Uilhajc6k| zsB=LDdAf7?n}Vkqtyao-?69E&om+TlIOR7X$-O<7#bpunH^f*;hlKx%9y7x~LXU&t zFX+waS@kj?0bM`&M0h&Eo#0p4%EJ^=B~8dw`uL@3xB1BkuJS%#EN!CWV~B63t_P2i z(v0d@I3ezgXlWQAHnNTynL*Mc+b)}<_Cr(9zG*a!sesyAPVSz;?CkBHTcrE#i2)vj-Ww{I0;$iM6q?gw?5Ub8}BQzgPykx_!5m<*bA;!N-LX+4VOtGyWr(|4$>8u?k{9>Nrvx{~<+L zt?23rCAfrWnZKd9fsd3p17V3`$o+hUU3i`kit3j56!JtidRCx1D{yZDu$va3Pp)A< 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