4.5: 章节公式回顾
- Page ID
- 205047
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)超几何分布
\(h(x)=\frac{\left(\begin{array}{l}{A} \\ {x}\end{array}\right)\left(\begin{array}{l}{N-A} \\ {n-x}\end{array}\right)}{\left(\begin{array}{l}{N} \\ {n}\end{array}\right)}\)
二项分布
\(X \sim B(n, p)\)意味着离散随机变量\(X\)具有二项式概率分布，包括\(n\)试验和成功概率\(p\)。
\(X =\)n 次独立试验的成功次数
\(n =\)独立试验的数量
\(X\)接受价值观\(x = 0, 1, 2, 3, ..., n\)
\(p =\)任何试验成功的概率
\(q =\)任何试验失败的概率
\(p + q = 1\)
\(q = 1 – p\)
的意思\(X\)是\(\mu = np\)。 的标准差\(X\)为\(\sigma=\sqrt{n p q}\)。
\[P(x)=\frac{n !}{x !(n-x) !} \cdot p^{x} q^{(n-x)}\nonumber\]
\(P(X)\)当任何一项\(n\)试验的\(X\)成功概率为时，试验成功的概率在哪里\(p\)。
几何分布
\(P(X=x)=p(1-p)^{x-1}\)
\(X \sim G(p)\)意味着离散随机变量\(X\)具有几何概率分布，在单次试验中具有成功概率\(p\)。
\(X =\)在第一次成功之前的独立试验次数
\(X\)接受价值观\(x = 1, 2, 3, ...\)
\(p =\)任何试验成功的概率
\(q =\)任何试验失败的概率\(p + q = 1\)
\(q = 1 – p\)
平均值是\(\mu = \frac{1}{p}\)。
标准差为\(\sigma=\sqrt{\frac{1-p}{p^{2}}}=\sqrt{\frac{1}{p}\left(\frac{1}{p}-1\right)}\)。
泊松分布
\(X \sim P(\mu )\)表示\(X\)具有泊松概率分布，其中\(X =\)感兴趣区间内的出现次数。
\(X\)接受价值观\(x = 0, 1, 2, 3, ...\)
通常给出平均值\(\mu\)或\(\lambda\)。
方差为\(\sigma ^2 = \mu\)，标准差为
\(\sigma=\sqrt{\mu}\)。
When\(P(\mu)\) 用于近似二项分布，\(\mu = np\)其中 n 表示独立试验的数量，\(p\)表示单次试验的成功概率。
\[P(x)=\frac{\mu^{x} e^{-\mu}}{x !}\nonumber\]