From b073dc7d0be649a76a95a4e182930e409ab65abe Mon Sep 17 00:00:00 2001 From: Didnelpsun <2675350965@qq.com> Date: Tue, 19 Oct 2021 23:42:48 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- .../random-events-and-probability.pdf | Bin 42889 -> 69551 bytes .../random-events-and-probability.tex | 47 ++++++++++++++++++ .../random-variables-and-distribution.pdf | Bin 76338 -> 84428 bytes .../random-variables-and-distribution.tex | 41 ++++++++++++++- 4 files changed, 87 insertions(+), 1 deletion(-) diff --git a/probability-theory-and-mathematical-statistics/exercise/1-random-events-and-probability/random-events-and-probability.pdf b/probability-theory-and-mathematical-statistics/exercise/1-random-events-and-probability/random-events-and-probability.pdf index e565354ae9015351bae71b96d35a5e523571e0cc..6f85973206a372fa3048d1b22b2654560c9e045e 100644 GIT binary patch literal 69551 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3238661..9eac873 100644 --- a/probability-theory-and-mathematical-statistics/exercise/1-random-events-and-probability/random-events-and-probability.tex +++ b/probability-theory-and-mathematical-statistics/exercise/1-random-events-and-probability/random-events-and-probability.tex @@ -34,6 +34,45 @@ \newpage \pagestyle{plain} \setcounter{page}{1} + +\section{排列组合} + +排列公式:$A_n^m=\dfrac{n!}{(n-m)!}$。 + +组合公式:$C_n^m=\dfrac{n!}{(n-m)!m!}$。 + +\subsection{捆绑法} + +要求某些元素必须在一起。 + +\textbf{例题:}$ABCDEF$六个人排队,要求$AB$必须在一起,问有多少种排法。 + +解:排法就是排列的问题。首先$AB$在一起,要么是$AB$要么是$BA$,也是一种排列,有$A_2^2$种。 + +然后将$AB$看作一个整体与$CDEF$进行排列,一共五个元素,进行全排列:$A_5^5$。 + +因为是按步骤来的,所以使用乘法:$A_2^2\cdot A_5^5=120$。 + +\subsection{插空法} + +要求某些元素不能相邻。 + +\textbf{例题:}要对6个唱歌和4个舞蹈节目进行排列,要求两个舞蹈节目不能相邻,求多少种排法。 + +解:由于是对舞蹈进行限制,所以对唱歌的序列没有特别的限制,第一步先对唱歌进行全排列$A_6^6$。 + +第二步对舞蹈进行排列,由于舞蹈之间不能相邻,所以舞蹈节目必然是在两个唱歌节目之间进行插孔排序的,而唱歌6个节目一共7个空,所以排列$A_7^4$。 + +由于是步骤,所以乘法:$A_6^6\cdot A_7^4$。 + +\subsection{插板法} + +与插空法类似,但是是组合进行归类,而不是排序。 + +\textbf{例题:}将8个完全相同的球放入三个不同的盒子,要求每个盒子至少有一个球,求一共有多少种放法。 + +解:相当于在七个空插入两个板。即$C_7^2=21$。 + \section{随机事件概率} 是基本事件关系的概率运算。 @@ -50,6 +89,14 @@ $ABC$都不发生就是$A$不发生且$B$不发生且$C$不发生,用式子表 \section{概率模型} +\subsection{古典概型} + +\textbf{例题:}设袋中黑白球各一个,有放回取球,每次取一个,直到两种颜色的球都取到为止。求取球次数恰为3的概率。 + +解:若是看到题目中的为止,很容易想到后面的几何概率分布。但是这与几何分布不同,因为几何分布要求的停止条件是同一个事件的前几次的失败和最后一次的成功,而这个题目要求是取到黑并且取到白,而不是取到黑或取到白这两个事件,如求取到黑时取求次数为3的概率,则是几何分布。 + +每次可能取到黑白两种,取三次,则事件总数为$2^3=8$。而取到第三次才取到黑白两色的,则有黑黑白、白白黑两种可能性,所以概率为$\dfrac{2}{8}=\dfrac{1}{4}$。 + \section{独立性} \textbf{例题:}射手对同一目标独立地进行4次射击。若至少命中一次的概率为$\dfrac{15}{16}$,则求该射手对同一目标独立地进行4次射击中至少没命中一次的概率。 diff --git a/probability-theory-and-mathematical-statistics/exercise/2-random-variables-and-distribution/random-variables-and-distribution.pdf b/probability-theory-and-mathematical-statistics/exercise/2-random-variables-and-distribution/random-variables-and-distribution.pdf index 92561efd29fdbd29d071527592afb74e28ee8337..6dca9f7f1fbf54e4ec83f0c4ac517ab6bbe1f63e 100644 GIT binary patch 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b/probability-theory-and-mathematical-statistics/exercise/2-random-variables-and-distribution/random-variables-and-distribution.tex @@ -41,10 +41,12 @@ \section{二项分布} +$P\{X=k\}=C_n^kp^k(1-p)^{n-k}$($k=0,1,\cdots,n$,$00$),$X\sim P(\lambda)$。 + \textbf{例题:}设一本书的各页印刷错误的个数$X$服从泊松分布。已知只有一个和只有两个印刷错误的页数相同,则随机抽查的4页中无印刷错误的概率$p$为? 解:$\because P\{X=1\}=P\{X=2\}$,$\therefore\dfrac{\lambda^1}{1!}e^{-\lambda}=\dfrac{\lambda^2}{2!}e^{-\lambda}$,$\lambda=2$。 @@ -62,6 +66,16 @@ $\therefore p=\left\{X\leqslant\dfrac{1}{2}\right\}=\int_0^\frac{1}{2}2x\,\textr \section{几何分布} +$P\{X=k\}=(1-p)^{k-1}p$($k=0,1,\cdots,n$,$00 \\ 0, \text{其他} @@ -77,6 +91,15 @@ $\therefore P\{Y=k\}=(k-1)\left(\dfrac{1}{8}\right)^2\cdot\left(\dfrac{7}{8}\rig \section{均匀分布} +$f(x)=\left\{\begin{array}{ll} + \dfrac{1}{b-a}, & a0$)且$P\{04\}=\dfrac{1}{2}$,求$X$的概率密度以及$P\{14\}=\dfrac{1}{2}$,4在其区间中点上,$\dfrac{a+b}{2}=4$。 @@ -147,14 +170,30 @@ $(X,Y)$联合概率=条件概率×边缘概率。 \section{指数分布} +$f(x)=\left\{\begin{array}{ll} + \lambda e^{-\lambda x}, & x>0 \\ + 0, & \text{其他} +\end{array}\right.$,$F(x)=\left\{\begin{array}{ll} + 1-e^{-\lambda x}, & x\geqslant0 \\ + 0, & x<0 \\ +\end{array}\right.$,$X\sim E(\lambda)$。 + \textbf{例题:}已知随机变量$X\sim E(1)$,$a$为常数且大于0,求$P\{X\leqslant a+1|X>a\}$。 解:$P\{X\leqslant a+1|X>a\}=\dfrac{P\{aa\}}=\dfrac{\int_a^{a+1}e^{-x}\,\textrm{d}x}{\int_a^{+\infty}e^{-x}\,\textrm{d}x}=1-\dfrac{1}{e}$。 也可以根据指数分布的无记忆性:$P\{X\leqslant a+1|X>a\}=1-P\{X>a+1|X>a\}=1-P\{X>1\}=P\{X\leqslant1\}=F(1)=1-\dfrac{1}{e}$。 +\textbf{例题:}随机变量$X$服从参数为1的指数分布,求$P\{3>X>2|X>1\}$。 + +已知$F(Xx)=e^{-\lambda x}$。且$2X>2|X>1\}=\dfrac{P\{3>X>2\}}{P\{X>1\}}=\dfrac{P\{X>2\}-P\{X>3\}}{P\{X>1\}}=\dfrac{e^{-2}-e^{-3}}{e^{-1}}=e^{-1}-e^{-2}$。 + \section{正态分布} +$f(x)=\dfrac{1}{\sqrt{2\pi\sigma}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$($-\infty0$),$X\sim N(\mu,\sigma^2)$。 + \textbf{例题:}已知随机变量$X\sim N(0,1)$,对给定的$\alpha$($0<\alpha>1$),数$\mu_\alpha$满足$P\{X>\mu_\alpha\}=\alpha$,若$P\{\vert X\vert\mu_\alpha\}=\alpha$即表示$\mu_\alpha$为标准正态分布的上$\alpha$分位点。