数学选修三
2023-04-09 10:03:27 18 举报AI智能生成
数学选修三是高中数学课程中的一门重要选修课程，主要涉及函数、数列、不等式、三角函数等内容。通过学习选修三，学生可以进一步加深对数学基本概念和原理的理解，提高解决实际问题的能力。此外，选修三还为学生提供了更多的数学工具和方法，如导数、极限等，为今后的学习和研究打下坚实的基础。总之，数学选修三是高中数学教育中不可或缺的一部分，对于培养学生的数学素养和综合素质具有重要意义。
作者其他创作
大纲/内容
计数原理
分类加法计数原理与分步乘法计数原理
分类加法计数原理
完成一件事有两类不同方案，在第1类方案中有<span class="equation-text" contenteditable="false" data-index="0" data-equation="m"><span></span><span></span></span>种不同的方法，在第2类方案中有<span class="equation-text" contenteditable="false" data-index="1" data-equation="n"><span></span><span></span></span>种不同的方法，<br>那么完成这件事共有<span class="equation-text" contenteditable="false" data-index="2" data-equation="N=m+n"><span></span><span></span></span>种不同的方法
分步乘法计数原理
完成一件事有两个步骤，做第1步有<span class="equation-text" contenteditable="false" data-index="0" data-equation="m"><span></span><span></span></span>种不同的方法，做第2步有<span class="equation-text" contenteditable="false" data-index="1" data-equation="n"><span></span><span></span></span>种不同的方法，<br>那么完成这件事共有<span class="equation-text" contenteditable="false" data-index="2" data-equation="N=m \times n"><span></span><span></span></span>种不同的方法
子集的个数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A=\left \{ a_1, a_2, \cdots a_n \right \} "><span></span><span></span></span>的子集有<span class="equation-text" contenteditable="false" data-index="1" data-equation="2^n"><span></span><span></span></span>个
排列与组合
排列
排列
从<span class="equation-text" contenteditable="false" data-index="0" data-equation="n"><span></span><span></span></span>个不同元素中取出<span class="equation-text" contenteditable="false" data-index="1" data-equation="m(m \le n)"><span></span><span></span></span>个元素，并按照一定的的顺序排成一列
排列数
排列数公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A_{n}^{m} = n(n-1)(n-2)\cdots (n-m+1) = \dfrac{n!}{m!}"><span></span><span></span></span>
全排列
<span class="equation-text" contenteditable="false" data-index="0" data-equation="A_{n}^{n} = n!"><span></span><span></span></span>
组合
组合
从<span class="equation-text" contenteditable="false" data-index="0" data-equation="n"><span></span><span></span></span>个不同元素中取出<span class="equation-text" contenteditable="false" data-index="1" data-equation="m(m \le n)"><span></span><span></span></span>个元素作为一组
组合数
组合数公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="C_{n}^{m} = \dfrac{A_{n}^{m}}{A_{m}^{m}} = \dfrac{n!}{m!(n-m)!} = \dbinom{n}{m} "><span></span><span></span></span>
组合数的两个性质
<span class="equation-text" contenteditable="false" data-index="0" data-equation="C_{n}^{m} = C_{n}^{n-m}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="C_{n+1}^{m} = C_{n}^{m} + C_{n}^{m-1}"><span></span><span></span></span>
二项式定理
二项式定理
二项式定理
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(a+b)^n = C_n^0 a^n + C_n^1 a^{n-1}b^1 + \cdots + C_n^k a^{n-k}b^k + \cdots + C_n^nb^n = \sum_{k=1}^{n}\dbinom{n}{k}a^{n-k}b^k ,\ n \in N^*"><span></span><span></span></span>
矩阵形式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(a+b)^{n}=\begin{bmatrix} a^0 & \cdots & a^n\end{bmatrix}\begin{bmatrix} C_{n}^{0} & & \\ & \ddots & \\ & & C_{n}^{n}\end{bmatrix}\begin{bmatrix} & & 1 \\ & \vdots & \\ 1 & &\end{bmatrix}\begin{bmatrix} b^0\\ \vdots\\ b^n\end{bmatrix}\quad n \in N^{+}"><span></span><span></span></span>
广义二项式定理
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(x+y)^{\alpha}=\sum_{k=0}^{\infty} \dbinom{\alpha}{k} x^{\alpha-k} y^{k}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\dbinom{\alpha}{k} =\dfrac{\alpha(\alpha-1) \ldots(\alpha-k+1)}{k!}=\dfrac{(\alpha)_{k}}{k!}"><span></span><span></span></span>
二项展开式
二项式系数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="C_n^k(k=0,1,2,\cdots,n)"><span></span><span></span></span>
通项
<span class="equation-text" contenteditable="false" data-index="0" data-equation="T_{k+1} = C_n^k a^{n-k}b^k"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(1+x)^n"><span></span><span></span></span>的展开
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(1+x)^n = C_n^0 + C_n^1x + C_n^2x^2 + \cdots + C_n^kx^k + \cdots + C_n^nx^n = \sum_{k=1}^{n}\dbinom{n}{k}x^k "><span></span><span></span></span>
二项式系数的性质
对称性
<span class="equation-text" contenteditable="false" data-index="0" data-equation="C_{n}^{m} = C_{n}^{n-m}"><span></span><span></span></span><br>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="r = \dfrac{n}{2}"><span></span><span></span></span>为函数<span class="equation-text" contenteditable="false" data-index="1" data-equation="f(r) = C_n^r"><span></span><span></span></span>图像的对称轴
增减性与最大值
<span class="equation-text" contenteditable="false" data-index="0" data-equation="n \equiv 2 (mod\ 0)"><span></span><span></span></span>：<span class="equation-text" contenteditable="false" data-index="1" data-equation="{\dbinom{n}{k}}_{max} = \dbinom{n}{\frac{n}{2}} "><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="n \equiv 2 (mod\ 1)"><span></span><span></span></span>：<span class="equation-text" contenteditable="false" data-index="1" data-equation="{\dbinom{n}{k}}_{max} = \dbinom{n}{\frac{n-1}{2}} = \dbinom{n}{\frac{n+1}{2}} "><span></span><span></span></span>
各二项式系数的和
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sum_{k=0}^{n} \dbinom{n}{k} = 2^n "><span></span><span></span></span>
拓展
多项式定理
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\dbinom{n}{n_{1}, n_{2}, \cdots, n_{t}}=\frac{n !}{n_{1} ! n_{2} ! \cdots n_{t} !} \ (n_{1,2,\cdots,t} \ge 0,\ \sum_{i=1}^{t}n_i = n)"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\dbinom{n}{n_{1}, n_{2}, \cdots, n_{t}}=\dbinom{n-1}{n_{1}-1, n_{2}, \cdots, n_{t}}+\dbinom{n-1}{n_{1}, n_{2}-1, \cdots, n_{t}}+\cdots +\dbinom{n-1}{n_{1}, n_{2}, \cdots, n_{t}-1}"><span></span><span></span></span>
自然数幂和公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(1+z)^a"><span></span><span></span></span>的展开
成对数据的统计分析
成对数据的统计相关性
变量的相关关系
相关关系
散点图
正相关/负相关
线性相关/非线性相关（曲线相关）
样本相关系数
样本相关系数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="r = \dfrac{\sum_{i=1}^{n} (x_i-\bar{x})(y_i-\bar{y})}{\sqrt{\sum_{i=1}^{n} (x_i-\bar{x})^2}\sqrt{\sum_{i=1}^{n} (y_i-\bar{y})^2}}"><span></span><span></span></span>
当<span class="equation-text" contenteditable="false" data-index="0" data-equation="|r|"><span></span><span></span></span>越接近于1时，成对样本数据的线性相关程度越强<br>当<span class="equation-text" data-index="1" data-equation="|r|" contenteditable="false"><span></span><span></span></span>越接近于0时，成对样本数据的线性相关程度越偌<br>
一元线性回归模型及其应用
一元线性回归模型
一元线性回归模型
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}Y = bx + a + e\\E(e) = 0,\ D(e) = \sigma^2\end{cases}"><span></span><span></span></span>
因变量/响应变量
<span class="equation-text" contenteditable="false" data-index="0" data-equation="Y"><span></span><span></span></span>
自变量/解释变量
<span class="equation-text" contenteditable="false" data-index="0" data-equation="x"><span></span><span></span></span>
随机误差（<span class="equation-text" contenteditable="false" data-index="0" data-equation="Y"><span></span><span></span></span>与<span class="equation-text" contenteditable="false" data-index="1" data-equation="bx+a"><span></span><span></span></span>之间）
<span class="equation-text" contenteditable="false" data-index="0" data-equation="e"><span></span><span></span></span>
一元线性回归模型参数的最小二乘估计
经验回归方程（经验回归函数/经验回归公式）
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\hat{y} = \hat{b}x + \hat{a}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\begin{cases}\hat{b} = \dfrac{\sum_{i=1}^{n} (x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^{n} (x_i-\bar{x})^2}\\\hat{a} = \bar{y} - \hat{b} \bar{x} \end{cases}"><span></span><span></span></span>
最小二乘法
最小二乘估计
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\hat{b}"><span></span><span></span></span>是<span class="equation-text" contenteditable="false" data-index="1" data-equation="b"><span></span><span></span></span>的最小二乘估计<br><span class="equation-text" contenteditable="false" data-index="2" data-equation="\hat{a}"><span></span><span></span></span>是<span class="equation-text" contenteditable="false" data-index="3" data-equation="a"><span></span><span></span></span>的最小二乘估计
观测值
预测值
残差
观测值<span class="equation-text" contenteditable="false" data-index="0" data-equation="-"><span></span><span></span></span>预测值
列联表与独立性检验
分类变量与列联表
分类变量
列联表
独立性检验
零假设/原假设
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\chi^2"><span></span><span></span></span>独立性检验（独立性检验）
随机变量及其分布
条件概率与全概率公式
条件概率
条件概率
在事件<span class="equation-text" contenteditable="false" data-index="0" data-equation="A"><span></span><span></span></span>发生的条件下，事件<span class="equation-text" contenteditable="false" data-index="1" data-equation="B"><span></span><span></span></span>发生的条件概率
<span class="equation-text" contenteditable="false" data-index="0" data-equation="P(B \mid A) = \dfrac{P(AB)}{P(A)}"><span></span><span></span></span>
概率的乘法公式
<span class="equation-text" contenteditable="false" data-index="0" data-equation="P(AB) = P(A)P(B \mid A)"><span></span><span></span></span>
全概率公式
全概率公式
设<span class="equation-text" contenteditable="false" data-index="0" data-equation="A_1, A_2, \cdots, A_n"><span></span><span></span></span>是一组两两互斥的事件，<span class="equation-text" contenteditable="false" data-index="1" data-equation="A_1 \bigcup A_2 \bigcup \cdots \bigcup A_n = \Omega"><span></span><span></span></span>，<br>且<span class="equation-text" contenteditable="false" data-index="2" data-equation="P(A_i) > 0,\ i = 1,2,\cdots ,n"><span></span><span></span></span>，则对任意的事件<span class="equation-text" contenteditable="false" data-index="3" data-equation="B\subseteq \Omega"><span></span><span></span></span>，有
<span class="equation-text" contenteditable="false" data-index="0" data-equation="P(B) = \sum_{i=1}^{n} P(A_i)P(B \mid A_i)"><span></span><span></span></span>
贝叶斯公式
设<span class="equation-text" contenteditable="false" data-index="0" data-equation="A_1, A_2, \cdots, A_n"><span></span><span></span></span>是一组两两互斥的事件，<span class="equation-text" contenteditable="false" data-index="1" data-equation="A_1 \bigcup A_2 \bigcup \cdots \bigcup A_n = \Omega"><span></span><span></span></span>，<br>且<span class="equation-text" contenteditable="false" data-index="2" data-equation="P(A_i) > 0,\ i = 1,2,\cdots ,n"><span></span><span></span></span>，则对任意的事件<span class="equation-text" contenteditable="false" data-index="3" data-equation="B\subseteq \Omega,\ P(B)>0"><span></span><span></span></span>，有
<span class="equation-text" contenteditable="false" data-index="0" data-equation="P(A_i \mid B) = \dfrac{P(A_i)P(B \mid A_i)}{P(B)} = \dfrac{P(A_i)P(B \mid A_i)}{\sum_{k=1}^{n} P(A_k)P(B \mid A_k)}"><span></span><span></span></span>
离散型随机变量及其分布列
随机变量
离散型随机变量
概率分布列（分布列）
<span class="equation-text" contenteditable="false" data-index="0" data-equation="P(X=x_i)=p_i,\ i=1,2,\cdots,n"><span></span><span></span></span>
两点分布（<span class="equation-text" contenteditable="false" data-index="0" data-equation="0-1"><span></span><span></span></span>分布）
离散型随机变量的数字特征
离散型随机变量的均值
随机变量<span class="equation-text" contenteditable="false" data-index="0" data-equation="X"><span></span><span></span></span>的均值或数学期望（期望）
<span class="equation-text" contenteditable="false" data-index="0" data-equation="E(X) = \sum_{i=1}^{n} x_ip_i"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="E(aX+b) = aE(X) + b"><span></span><span></span></span>
离散型随机变量的方差
方差
<span class="equation-text" contenteditable="false" data-index="0" data-equation="D(X) = Var(X) = \sum_{i=1}^{n} (x_i - E(X))^2p_i = \sum_{i=1}^{n} x_i^2p_i - (E(X))^2"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="D(aX+b) = a^2D(X)"><span></span><span></span></span>
准差
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sigma(X) = \sqrt{D(X)}"><span></span><span></span></span>
二项分布与超几何分布
二项分布
伯努利试验
只包含两个可能结果的试验
<span class="equation-text" contenteditable="false" data-index="0" data-equation="n"><span></span><span></span></span>重伯努利试验
一个伯努利试验独立地重复<span class="equation-text" contenteditable="false" data-index="0" data-equation="n"><span></span><span></span></span>次所组成的随机试验
二项分布
在<span class="equation-text" contenteditable="false" data-index="0" data-equation="n"><span></span><span></span></span>重伯努利试验中，设每次试验中事件<span class="equation-text" contenteditable="false" data-index="1" data-equation="A"><span></span><span></span></span>发生的概率为<span class="equation-text" contenteditable="false" data-index="2" data-equation="p(0<p<1)"><span></span><span></span></span>，用<span class="equation-text" contenteditable="false" data-index="3" data-equation="X"><span></span><span></span></span>表示事件<span class="equation-text" contenteditable="false" data-index="4" data-equation="A"><span></span><span></span></span>发生的次数，则<span class="equation-text" contenteditable="false" data-index="5" data-equation="X"><span></span><span></span></span>的分布列为
<span class="equation-text" contenteditable="false" data-index="0" data-equation="P(X=k) = C_{n}^{k}p^k(1-p)^{n-k},\ k=0,1,2,\cdots,n"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sum_{k=0}^{n} P(X=k) = 1"><span></span><span></span></span>
随机变量<span class="equation-text" contenteditable="false" data-index="0" data-equation="X"><span></span><span></span></span>服从二项分布，记作<span class="equation-text" contenteditable="false" data-index="1" data-equation="X \sim B(n,p)"><span></span><span></span></span>
二项分布的均值和方差
<span class="equation-text" contenteditable="false" data-index="0" data-equation="X \sim B(n,p) \Longrightarrow E(X)=np,\ D(x) = np(1-p)"><span></span><span></span></span>
超几何分布
超几何分布
<span class="equation-text" contenteditable="false" data-index="0" data-equation="P(X=k) = \dfrac{C_M^kC_{N-M}^{k}}{C_N^n},\ k=m,m+1,\cdots,r"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="n,N,M \in N^* \quad M\le N, n\le N \quad m=\max \left \{ 0,n-N+M \right \}, r = \min \left \{ n,M \right \}"><span></span><span></span></span>
超几何分布的均值
<span class="equation-text" contenteditable="false" data-index="0" data-equation="E(X) = np"><span></span><span></span></span>
二项分布的性质
<span class="equation-text" contenteditable="false" data-index="0" data-equation="k<(n+1)p"><span></span><span></span></span>：<span class="equation-text" contenteditable="false" data-index="1" data-equation="p_k>p_{k-1}"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="2" data-equation="p_k"><span></span><span></span></span>随<span class="equation-text" contenteditable="false" data-index="3" data-equation="k"><span></span><span></span></span>值的增加而增加
<span class="equation-text" contenteditable="false" data-index="0" data-equation="k>(n+1)p"><span></span><span></span></span>：<span class="equation-text" contenteditable="false" data-index="1" data-equation="p_k<p_{k-1}"><span></span><span></span></span>，<span class="equation-text" contenteditable="false" data-index="2" data-equation="p_k"><span></span><span></span></span>随<span class="equation-text" contenteditable="false" data-index="3" data-equation="k"><span></span><span></span></span>值的增加而减小
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(n+1)p"><span></span><span></span></span>为正整数：当<span class="equation-text" contenteditable="false" data-index="1" data-equation="k=(n+1)p"><span></span><span></span></span>时，<span class="equation-text" contenteditable="false" data-index="2" data-equation="p_k=p_{k-1}"><span></span><span></span></span>为最大值
<span class="equation-text" contenteditable="false" data-index="0" data-equation="(n+1)p"><span></span><span></span></span>为非整数：当<span class="equation-text" contenteditable="false" data-index="1" data-equation="k=[(n+1)p]"><span></span><span></span></span>时，<span class="equation-text" contenteditable="false" data-index="2" data-equation="p_k"><span></span><span></span></span>是唯一的最大值
正态分布
连续型随随机变量
正态密度函数
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\mu \in R,\ \sigma>0"><span></span><span></span></span>
正态密度曲线（正态曲线）
正态分布（高斯分布）
<span class="equation-text" contenteditable="false" data-index="0" data-equation="X \sim N(\mu, \sigma^2)"><span></span><span></span></span>
标准正态分布
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\mu = 0,\ \sigma = 1"><span></span><span></span></span>
正态分布下的概率
正态曲线的特点
曲线是单峰的，关于直线<span class="equation-text" contenteditable="false" data-index="0" data-equation="x=\mu"><span></span><span></span></span>对称
曲线在<span class="equation-text" contenteditable="false" data-index="0" data-equation="x=\mu"><span></span><span></span></span>处达到峰值<span class="equation-text" contenteditable="false" data-index="1" data-equation="\dfrac{1}{\sigma\sqrt{2\pi}}"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\lim_{x \to \infty} f(x) = 0"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="\sigma"><span></span><span></span></span>越大，曲线越“瘦高”，表示随机变量<span class="equation-text" contenteditable="false" data-index="1" data-equation="X"><span></span><span></span></span>的分布比较集中<br><span class="equation-text" contenteditable="false" data-index="2" data-equation="\sigma"><span></span><span></span></span>越小，曲线越“矮胖”，表示随机变量<span class="equation-text" contenteditable="false" data-index="3" data-equation="X"><span></span><span></span></span>的分布比较分散
正态分布的均值和方差
<span class="equation-text" contenteditable="false" data-index="0" data-equation="E(x) =\mu,\ D(X) = \sigma^2"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="3\sigma"><span></span><span></span></span>原则
<span class="equation-text" contenteditable="false" data-index="0" data-equation="P(\mu - \sigma \le X \le \mu + \sigma) \approx 0.6827"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="P(\mu - 2\sigma \le X \le \mu + 2\sigma) \approx 0.9545"><span></span><span></span></span>
<span class="equation-text" contenteditable="false" data-index="0" data-equation="P(\mu - 3\sigma \le X \le \mu + 3\sigma) \approx 0.9973"><span></span><span></span></span>
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