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概率论与数理统计第4章随机变量的数字特征

2026-05-29T10:00:00+00:00

章节概览

本章主要介绍随机变量的各种数字特征（Numerical Characteristics），包括：

4.1 随机变量的期望 (Expectation of a Random Variable)
4.2 方差 (Variance)
4.3 常见分布的数字特征 (Characteristics of Common Distributions)
4.4 切比雪夫不等式 (Chebyshev’s Inequality)
4.5 协方差与相关系数 (Covariance and Correlation)
4.6 矩与协方差矩阵 (Moments and Covariance Matrices)

4.1 随机变量的期望 (Expectation of a Random Variable)

背景说明

随机变量的分布函数（CDF）完整地描述了其统计规律性，但有时我们需要了解随机变量的某些特征值，如平均值或期望值，这些特征值可以让我们在不描述整个分布的情况下了解变量的中心位置。

4.1.1 离散型随机变量的期望

定义 4.1.1 (Definition 4.1.1)

设 $X$ 是离散型随机变量，其概率分布律（PF, Probability Function）为：

\[P(X = x_k) = p_k, \quad k = 1, 2, \cdots\]

若级数 $\sum_{k=1}^{\infty} x_k p_k$ 绝对收敛，则称 $X$ 的期望（Expectation/Mean/Expected Value）存在，定义为：

\[E(X) = \sum_{k=1}^{\infty} x_k p_k\]

注释 (Remark)：不仅要求级数收敛，还要求绝对收敛（absolutely convergent），因为非绝对收敛的无穷级数重排后可能得到不同的和。绝对收敛保证了期望的唯一性。

4.1.2 连续型随机变量的期望

定义 4.1.2 (Definition 4.1.2)

设 $X$ 是连续型随机变量，其概率密度函数（PDF, Probability Density Function）为 $f(x)$，若积分 $\int_{-\infty}^{\infty} x f(x) dx$ 绝对收敛，则称 $X$ 的期望存在，定义为：

\[E(X) = \int_{-\infty}^{\infty} x f(x) dx\]

4.1.3 随机变量函数的期望

定理 4.1.1 (Theorem 4.1.1)

设 $X$ 是随机变量，$Y = r(X)$ 是 $X$ 的函数：

(1) 离散情形： 若 $X$ 的分布律为 $P(X = x_k) = p_k$，且 $\sum_{k=1}^{\infty} r(x_k) p_k$ 绝对收敛，则：

\[E(Y) = \sum_{k=1}^{\infty} r(x_k) p_k\]

(2) 连续情形： 若 $X$ 的密度函数为 $f(x)$，且 $\int_{-\infty}^{\infty} r(x) f(x) dx$ 绝对收敛，则：

\[E(Y) = \int_{-\infty}^{\infty} r(x) f(x) dx\]

4.1.4 多维随机变量函数的期望

定理 4.1.2 (Theorem 4.1.2)

设 $X_1, \cdots, X_n$ 是具有联合概率密度函数 $f(x_1, \cdots, x_n)$ 的随机变量，$Y = r(X_1, \cdots, X_n)$，则：

\[E(Y) = \int_{-\infty}^{\infty} \cdots \int_{-\infty}^{\infty} r(x_1, \cdots, x_n) f(x_1, \cdots, x_n) dx_1 \cdots dx_n\]

4.1.5 期望的性质

定理 4.1.3 (Theorem 4.1.3) - 线性性质

若 $Y = aX + b$，其中 $a, b$ 为有限常数，则：

\[E(Y) = aE(X) + b\]

注释：一般情况下，$E(XY) \neq E(X)E(Y)$。

定理 4.1.4 (Theorem 4.1.4) - 保序性

若存在常数 $a$ 使得 $P(X \geq a) = 1$，则 $E(X) \geq a$
若存在常数 $b$ 使得 $P(X \leq b) = 1$，则 $E(X) \leq b$

定理 4.1.5 (Theorem 4.1.5) - 和的期望

若 $X_1, \cdots, X_n$ 是 $n$ 个随机变量，且每个期望 $E(X_i)$ 都存在 $(i = 1, \cdots, n)$，则：

\[E(X_1 + \cdots + X_n) = E(X_1) + \cdots + E(X_n)\]

注释：即使 $X_1, \cdots, X_n$ 不独立，此性质也成立。

定理 4.1.6 (Theorem 4.1.6) - 独立随机变量积的期望

若 $X_1, \cdots, X_n$ 是 $n$ 个相互独立的随机变量，且每个期望 $E(X_i)$ 都存在 $(i = 1, \cdots, n)$，则：

\[E(X_1 X_2 \cdots X_n) = E(X_1) \times E(X_2) \times \cdots \times E(X_n)\]

4.2 方差 (Variance)

4.2.1 方差的定义

定义 4.2.1 (Definition 4.2.1)

设 $X$ 是具有有限均值 $\mu = E(X)$ 的随机变量，$X$ 的方差（Variance）定义为：

\[\sigma_X^2 = \text{Var}(X) = E[(X - \mu)^2]\]

若 $X$ 的均值无穷大或不存在，则称 $\text{Var}(X)$ 不存在
标准差（Standard Deviation）$\sigma_X$ 是方差的非负平方根

方差的意义：方差描述了随机变量取值与其期望之间的偏离程度。取值越集中，方差越小；取值越分散，方差越大。方差是衡量随机变量离散程度的指标。

4.2.2 方差的计算公式

定理 4.2.1 (Theorem 4.2.1)

对任意随机变量 $X$：

\[\text{Var}(X) = E(X^2) - [E(X)]^2\]

4.2.3 方差的性质

定理 4.2.2 (Theorem 4.2.2)

对每个随机变量 $X$，$\text{Var}(X) \geq 0$。若 $X$ 是有界随机变量，则 $\text{Var}(X)$ 必定存在且有限。

定理 4.2.3 (Theorem 4.2.3)

$\text{Var}(X) = 0$ 当且仅当存在常数 $c = E(X)$ 使得 $P(X = c) = 1$。

定理 4.2.4 (Theorem 4.2.4) - 方差的展开

若 $X$ 和 $Y$ 是满足 $\text{Var}(X) < \infty$ 和 $\text{Var}(Y) < \infty$ 的随机变量，$a, b, c$ 为常数，则：

\[\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2[E(XY) - E(X)E(Y)]\] \[\text{Var}(aX + bY + c) = a^2\text{Var}(X) + b^2\text{Var}(Y) + 2ab[E(XY) - E(X)E(Y)]\]

定理 4.2.5 (Theorem 4.2.5) - 独立随机变量线性组合的方差

若 $X_1, \cdots, X_n$ 是具有有限均值的独立随机变量，$a_1, \cdots, a_n$ 和 $b$ 是任意常数，则：

\[\text{Var}(a_1 X_1 + \cdots + a_n X_n + b) = a_1^2 \text{Var}(X_1) + \cdots + a_n^2 \text{Var}(X_n)\]

4.3 常见分布的数字特征 (Characteristics of Common Distributions)

4.3.1 伯努利分布 (Bernoulli Distribution)

定理 4.3.1 (Theorem 4.3.1)

若随机变量 $X$ 服从参数为 $p$ $(0 \leq p \leq 1)$ 的伯努利分布，即 $X$ 只能取 0 和 1，且：

\[P(X = 1) = p, \quad P(X = 0) = 1 - p\]

则：

\[E(X) = p, \quad \text{Var}(X) = p(1-p)\]

伯努利试验 (Bernoulli Trials)：若有限或无限序列 $X_1, X_2, \cdots$ 中的随机变量相互独立同分布（IID），且每个 $X_i$ 服从参数为 $p$ 的伯努利分布，则称 $X_1, X_2, \cdots$ 为参数为 $p$ 的伯努利试验。

4.3.2 二项分布 (Binomial Distribution)

定理 4.3.2 (Theorem 4.3.2)

若随机变量 $X_1, \cdots, X_n$ 构成 $n$ 次参数为 $p$ 的伯努利试验，且 $X = X_1 + \cdots + X_n$，则 $X$ 服从参数为 $n$ 和 $p$ 的二项分布 $B(n, p)$，其分布律为：

\[P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}, \quad k = 0, 1, \cdots, n\]

则：

\[E(X) = np, \quad \text{Var}(X) = np(1-p)\]

4.3.3 泊松分布 (Poisson Distribution)

定理 4.3.3 (Theorem 4.3.3)

若 $X \sim \text{Poisson}(\lambda)$，则：

\[E(X) = \text{Var}(X) = \lambda\]

分布律为：

\[P(X = k) = \frac{\lambda^k e^{-\lambda}}{k!}, \quad k = 0, 1, 2, \cdots\]

4.3.4 超几何分布 (Hypergeometric Distribution)

定理 4.3.4 (Theorem 4.3.4)

设 $X \sim H(N, m, n)$，其中参数 $N, m, n$ 均为正数，则：

\[E(X) = \frac{nm}{N}\] \[\text{Var}(X) = \frac{nm(N-m)(N-n)}{N^2(N-1)}\]

4.3.5 负二项分布 (Negative Binomial Distribution)

定理 4.3.5 (Theorem 4.3.5)

若 $X$ 服从参数为 $r$ 和 $p$ 的负二项分布，则：

\[E(X) = \frac{r}{p}, \quad \text{Var}(X) = \frac{r(1-p)}{p^2}\]

4.3.6 均匀分布 (Uniform Distribution)

定理 4.3.6 (Theorem 4.3.6)

设 $X \sim U[a, b]$，则：

\[E(X) = \frac{a+b}{2}, \quad \text{Var}(X) = \frac{(b-a)^2}{12}\]

概率密度函数为：

\[f(x) = \begin{cases} \frac{1}{b-a}, & a \leq x \leq b \\ 0, & \text{其他} \end{cases}\]

4.3.7 指数分布 (Exponential Distribution)

定理 4.3.7 (Theorem 4.3.7)

设 $X \sim \text{Exp}(\lambda)$，则：

\[E(X) = \frac{1}{\lambda}, \quad \text{Var}(X) = \frac{1}{\lambda^2}\]

概率密度函数为：

\[f(x) = \begin{cases} \lambda e^{-\lambda x}, & x > 0 \\ 0, & x \leq 0 \end{cases}\]

4.3.8 正态分布 (Normal Distribution)

定理 4.3.8 (Theorem 4.3.8)

设 $X \sim N(\mu, \sigma^2)$，则：

\[E(X) = \mu, \quad \text{Var}(X) = \sigma^2\]

概率密度函数为：

\[f(x) = \frac{1}{\sqrt{2\pi}\sigma} e^{-\frac{(x-\mu)^2}{2\sigma^2}}, \quad -\infty < x < \infty\]

其中 $-\infty < \mu < \infty$ 和 $\sigma > 0$ 是两个参数。

标准化：令 $Y = \frac{X - \mu}{\sigma}$，则 $Y \sim N(0, 1)$（标准正态分布）。

重要说明：正态分布的两个参数恰好是其均值和方差，正态分布完全由其均值和方差确定。

4.3.9 Gamma分布 (Gamma Distribution)

定理 4.3.9 (Theorem 4.3.9)

设 $X \sim \text{Gamma}(\alpha, \beta)$，则：

\[E(X) = \alpha\beta, \quad \text{Var}(X) = \alpha\beta^2\]

备注 4.3.1 (Remark 4.3.1)

特别地，当 $\alpha = \frac{n}{2}$，$\beta = 2$ 时，Gamma分布退化为自由度为 $n$ 的 卡方分布（Chi-squared Distribution）$\chi^2(n)$：

若 $X \sim \chi^2(n)$，则 $E(X) = n$，$\text{Var}(X) = 2n$

4.4 切比雪夫不等式 (Chebyshev’s Inequality)

4.4.1 切比雪夫不等式

定理 4.4.1 (Theorem 4.4.1)

设 $X$ 是具有均值 $E(X)$ 和方差 $\text{Var}(X) = \sigma^2$ 的随机变量，则对任意 $\varepsilon > 0$：

\[P(|X - E(X)| \geq \varepsilon) \leq \frac{\text{Var}(X)}{\varepsilon^2} = \frac{\sigma^2}{\varepsilon^2}\]

等价形式：

\[P(|X - E(X)| < \varepsilon) \geq 1 - \frac{\text{Var}(X)}{\varepsilon^2}\]

或令 $k = \frac{\varepsilon}{\sigma}$：

\[P(|X - E(X)| \geq k\sigma) \leq \frac{1}{k^2}\]

4.4.2 证明思路

注释 4.4.1 (Remark 4.4.1)

切比雪夫不等式提供了一种在不知道 $X$ 具体分布的情况下近似概率的方法。

注释 4.4.2 (Remark 4.4.2)

由于切比雪夫不等式不对分布形状做任何假设，这是一个缺点。虽然不等式成立，但它过于一般化，实用性有限，因此主要用于理论研究。切比雪夫的结果虽然正确，但与实际值相比非常保守。

4.5 协方差与相关系数 (Covariance and Correlation)

4.5.1 协方差的定义

定义 4.5.1 (Definition 4.5.1)

设 $X$ 和 $Y$ 是具有有限均值的随机变量，$E(X) = \mu_X$，$E(Y) = \mu_Y$。$X$ 和 $Y$ 的协方差（Covariance）定义为：

\[\text{Cov}(X, Y) = E[(X - \mu_X)(Y - \mu_Y)] = E(XY) - E(X)E(Y)\]

特别地，当 $X = Y$ 时，$\text{Cov}(X, X) = \text{Var}(X)$。

协方差的意义：协方差是衡量两个随机变量之间关联性质的指标，它只描述两个随机变量之间的线性关系。

4.5.2 协方差与方差的关系

定理 4.5.1 (Theorem 4.5.1)

若 $X$ 和 $Y$ 是满足 $\text{Var}(X) < \infty$ 和 $\text{Var}(Y) < \infty$ 的随机变量，$a, b, c$ 为常数，则：

\[\text{Var}(X + Y) = \text{Var}(X) + \text{Var}(Y) + 2\text{Cov}(X, Y)\] \[\text{Var}(aX + bY + c) = a^2\text{Var}(X) + b^2\text{Var}(Y) + 2ab\text{Cov}(X, Y)\]

4.5.3 协方差的性质

命题 4.5.1 (Proposition 4.5.1)

协方差具有以下性质：对随机变量 $X, Y, Z$ 和常数 $a, b$：

对称性：$\text{Cov}(X, Y) = \text{Cov}(Y, X)$
齐次性：$\text{Cov}(aX, bY) = ab\text{Cov}(X, Y)$
可加性：$\text{Cov}(X + Y, Z) = \text{Cov}(X, Z) + \text{Cov}(Y, Z)$

4.5.4 相关系数的定义

定义 4.5.2 (Definition 4.5.2)

设 $X$ 和 $Y$ 是具有有限方差 $\sigma_X^2$ 和 $\sigma_Y^2$ 的随机变量，则 $X$ 和 $Y$ 的相关系数（Correlation Coefficient）定义为：

\[\rho_{X,Y} = \frac{\text{Cov}(X, Y)}{\sigma_X \sigma_Y} = \frac{\text{Cov}(X, Y)}{\sqrt{\text{Var}(X)}\sqrt{\text{Var}(Y)}}\]

备注 4.5.1 (Remark 4.5.1)

令 $X^* = \frac{X - \mu_X}{\sigma_X}$，$Y^* = \frac{Y - \mu_Y}{\sigma_Y}$，则 $E(X^*) = E(Y^*) = 0$，$\text{Var}(X^*) = \text{Var}(Y^*) = 1$，$X^*$ 和 $Y^*$ 分别称为 $X$ 和 $Y$ 的标准化变量（Standardized Variables）。

\[\rho_{X,Y} = \text{Cov}(X^*, Y^*)\]

$\rho_{X,Y}$ 是无量纲的（free of units）。

4.5.5 相关系数的性质

定理 4.5.2 (Theorem 4.5.2)

4.5.6 相关性的分类

定义 4.5.3 (Definition 4.5.3)

若 $\text{Cov}(X, Y) > 0$，称 $X$ 和 $Y$ 正相关（Positive Correlated）
若 $\text{Cov}(X, Y) < 0$，称 $X$ 和 $Y$ 负相关（Negative Correlated）
若 $\text{Cov}(X, Y) = 0$，称 $X$ 和 $Y$ 不相关（Uncorrelated）

4.5.7 独立性与相关性的关系

备注 4.5.2 (Remark 4.5.2)

若 $X$ 和 $Y$ 不相关，意味着 $X$ 和 $Y$ 之间没有线性关系，但可能存在其他关系
由定理 4.1.6，若 $X$ 和 $Y$ 独立，则 $E(XY) = E(X)E(Y)$，$\text{Cov}(X, Y) = 0$
即：独立性蕴含不相关性
等价地，若 $\text{Cov}(X, Y) \neq 0$ 或 $\rho_{X,Y} \neq 0$，则 $X$ 和 $Y$ 不独立
然而，若 $X$ 和 $Y$ 不相关，它们可能是依赖的也可能是独立的
不相关的随机变量不一定独立！

4.5.8 二元正态分布的相关性

命题 4.5.3 (Proposition 4.5.3)

设 $(X, Y)$ 服从二元正态分布 $N(\mu_1, \mu_2, \sigma_1^2, \sigma_2^2, \rho)$，则参数 $\rho$ 恰好是 $X$ 和 $Y$ 的相关系数。

重要性质：设 $(X, Y)$ 服从二元正态分布，则 $X$ 和 $Y$ 独立当且仅当它们不相关（即 $\rho = 0$）。

4.6 矩与协方差矩阵 (Moments and Covariance Matrices)

4.6.1 矩的定义

定义 4.6.1 (Definition 4.6.1)

设 $X$ 是随机变量，且 $E(X^k)$ 存在，$k = 1, 2, \cdots$：

$E(X^k)$ 称为 $X$ 的 $k$ 阶原点矩（$k$-th Moment about the Origin）
$E\{[X - E(X)]^k\}$（$k = 2, 3, \cdots$）若存在，称为 $X$ 的 $k$ 阶中心矩（$k$-th Central Moment）
$E\{[X - E(X)]^k [Y - E(Y)]^l\}$（$k, l = 1, 2, \cdots$）若存在，称为 $X$ 和 $Y$ 的 $k+l$ 阶混合中心矩（Mixed Central Moment of Order $k+l$）

重要说明：

均值 $E(X)$ 是 $X$ 的一阶原点矩
方差 $\text{Var}(X)$ 是 $X$ 的二阶中心矩
协方差 $\text{Cov}(X, Y)$ 是 $X$ 和 $Y$ 的二元二阶混合中心矩

4.6.2 协方差矩阵的定义

定义 4.6.2 (Definition 4.6.2)

设 $(X_1, X_2)$ 的二阶中心矩存在，记为：

\[c_{11} = \text{Var}(X_1), \quad c_{22} = \text{Var}(X_2), \quad c_{12} = c_{21} = \text{Cov}(X_1, X_2)\]

则矩阵：

\[C = \begin{pmatrix} c_{11} & c_{12} \\ c_{21} & c_{22} \end{pmatrix}\]

定义为 $(X_1, X_2)$ 的协方差矩阵（Covariance Matrix）。

推广到 $n$ 维：

$(X_1, X_2, \cdots, X_n)$ 的协方差矩阵定义为：

\[C = \begin{pmatrix} c_{11} & c_{12} & \cdots & c_{1n} \\ c_{21} & c_{22} & \cdots & c_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ c_{n1} & c_{n2} & \cdots & c_{nn} \end{pmatrix}\]

其中 $c_{ij} = \text{Cov}(X_i, X_j)$，$i, j = 1, 2, \cdots, n$。

由于 $c_{ij} = c_{ji}$（$i \neq j$），协方差矩阵是对称矩阵（Symmetric Matrix）。

4.6.3 二元正态分布的协方差矩阵表示

设 $(X_1, X_2) \sim N(\mu_1, \mu_2, \sigma_1^2, \sigma_2^2, \rho)$，则协方差矩阵为：

\[C = \begin{pmatrix} \sigma_1^2 & \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma_2^2 \end{pmatrix}\]

4.6.4 非退化情况的概率密度函数

若 $$

\neq 0$，则$C$的逆矩阵存在，$(X, Y)$称为**非退化的**（Non-degenerate）。若$

= 0$，$(X, Y)$$ 是退化的（Degenerate），没有密度函数。

对于非退化情况，$(X_1, X_2)$ 的概率密度函数为：

\[f(x_1, x_2) = \frac{1}{2\pi|C|^{1/2}} \exp\left\{-\frac{1}{2}(X - \mu)^T C^{-1} (X - \mu)\right\}\]

其中 $X = \begin{pmatrix} x_1 \\ x_2 \end{pmatrix}$，$\mu = \begin{pmatrix} \mu_1 \\ \mu_2 \end{pmatrix}$，$$

$和$C^{-1}$分别是$C$的行列式和逆矩阵，$(X - \mu)^T$是$(X - \mu)$$ 的转置。

4.6.5 $n$ 维正态分布

上述结果可推广到 $n$ 维正态随机向量 $(X_1, X_2, \cdots, X_n)$，概率密度函数为：

\[f(x_1, \cdots, x_n) = \frac{1}{(2\pi)^{n/2}|C|^{1/2}} \exp\left\{-\frac{1}{2}(X - \mu)^T C^{-1} (X - \mu)\right\}\]

4.6.6 $n$ 维正态分布的性质

$n$ 维正态随机向量 $(X_1, \cdots, X_n)$ 具有以下性质：

边缘分布：给定 $n$ 维正态随机向量 $(X_1, \cdots, X_n)$，其所有分量 $X_i$（$i = 1, \cdots, n$）都是正态随机变量；反之，若 $X_i$（$i = 1, \cdots, n$）是相互独立的正态随机变量，则 $(X_1, \cdots, X_n)$ 是 $n$ 维正态随机向量。
线性组合：$(X_1, \cdots, X_n)$ 服从 $n$ 维正态分布当且仅当其任意线性组合 $a_1 X_1 + \cdots + a_n X_n$ 服从正态分布（$a_1, \cdots, a_n$ 不全为零）。
线性变换的不变性：若 $(X_1, \cdots, X_n)$ 服从 $n$ 维正态分布，$Y_1, Y_2, \cdots, Y_m$ 是 $X_i$（$i = 1, \cdots, n$）的线性函数（系数不全为零），则 $(Y_1, Y_2, \cdots, Y_m)$ 服从 $m$ 维正态分布。

注释：这表明正态分布在线性变换下保持不变。上述 $m$ 维正态分布可能是退化的，没有密度函数。
独立性与不相关性等价：若 $(X_1, \cdots, X_n)$ 服从 $n$ 维正态分布，则 $X_1, \cdots, X_n$ 相互独立当且仅当它们两两不相关。

4.6.7 正态随机向量的线性变换

定理 4.6.2 (Theorem 4.6.2)

若 $(X_1, X_2) \sim N(\mu_1, \mu_2, \sigma_1^2, \sigma_2^2, \rho)$，则：

\[aX_1 + bX_2 \sim N(a\mu_1 + b\mu_2, a^2\sigma_1^2 + b^2\sigma_2^2 + 2ab\rho\sigma_1\sigma_2)\]

关键术语中英对照表

中文	英文
期望/均值	Expectation/Mean/Expected Value
方差	Variance
标准差	Standard Deviation
概率分布律	Probability Function (PF)
概率密度函数	Probability Density Function (PDF)
累积分布函数	Cumulative Distribution Function (CDF)
伯努利分布	Bernoulli Distribution
二项分布	Binomial Distribution
泊松分布	Poisson Distribution
超几何分布	Hypergeometric Distribution
负二项分布	Negative Binomial Distribution
均匀分布	Uniform Distribution
指数分布	Exponential Distribution
正态分布/高斯分布	Normal/Gaussian Distribution
Gamma分布	Gamma Distribution
卡方分布	Chi-squared Distribution
切比雪夫不等式	Chebyshev’s Inequality
协方差	Covariance
相关系数	Correlation Coefficient
正相关	Positive Correlated
负相关	Negative Correlated
不相关	Uncorrelated
独立	Independent
矩	Moment
原点矩	Moment about the Origin
中心矩	Central Moment
混合矩	Mixed Moment
协方差矩阵	Covariance Matrix
对称矩阵	Symmetric Matrix
退化/非退化	Degenerate/Non-degenerate
标准化变量	Standardized Variables
二元正态分布	Bivariate Normal Distribution
$n$ 维正态分布	$n$-dimensional Normal Distribution
线性变换	Linear Transformation
独立同分布	Independent and Identically Distributed (IID)

a post with plotly.js

2025-03-26T14:24:00+00:00

This is an example post with some plotly code.

```plotly
{
  "data": [
    {
      "x": [1, 2, 3, 4],
      "y": [10, 15, 13, 17],
      "type": "scatter"
    },
    {
      "x": [1, 2, 3, 4],
      "y": [16, 5, 11, 9],
      "type": "scatter"
    }
  ]
}
```

Which generates:

{
  "data": [
    {
      "x": [1, 2, 3, 4],
      "y": [10, 15, 13, 17],
      "type": "scatter"
    },
    {
      "x": [1, 2, 3, 4],
      "y": [16, 5, 11, 9],
      "type": "scatter"
    }
  ]
}

Also another example chart.

```plotly
{
  "data": [
    {
      "x": [1, 2, 3, 4],
      "y": [10, 15, 13, 17],
      "mode": "markers"
    },
    {
      "x": [2, 3, 4, 5],
      "y": [16, 5, 11, 9],
      "mode": "lines"
    },
    {
      "x": [1, 2, 3, 4],
      "y": [12, 9, 15, 12],
      "mode": "lines+markers"
    }
  ],
  "layout": {
    "title": {
      "text": "Line and Scatter Plot"
    }
  }
}
```

This is how it looks like:

{
  "data": [
    {
      "x": [1, 2, 3, 4],
      "y": [10, 15, 13, 17],
      "mode": "markers"
    },
    {
      "x": [2, 3, 4, 5],
      "y": [16, 5, 11, 9],
      "mode": "lines"
    },
    {
      "x": [1, 2, 3, 4],
      "y": [12, 9, 15, 12],
      "mode": "lines+markers"
    }
  ],
  "layout": {
    "title": {
      "text": "Line and Scatter Plot"
    }
  }
}

a post with image galleries

2024-12-04T01:59:00+00:00

The images in this post are all zoomable, arranged into different mini-galleries using different libraries.

Lightbox2

PhotoSwipe

Spotlight JS

Venobox

a post with tabs

2024-05-01T00:32:13+00:00

This is how a post with tabs looks like. Note that the tabs could be used for different purposes, not only for code.

First tabs

To add tabs, use the following syntax:

{% tabs group-name %}

{% tab group-name tab-name-1 %}

Content 1

{% endtab %}

{% tab group-name tab-name-2 %}

Content 2

{% endtab %}

{% endtabs %}

With this you can generate visualizations like:

```
var_dump('hello');
```
```
console.log("hello");
```
```
pputs 'hello'
```

Another example

yaml
json

```
hello:
  - "whatsup"
  - "hi"
```
```
{
  "hello": ["whatsup", "hi"]
}
```

Tabs for something else

Regular text
A quote
Hipster list
- brunch
- fixie
- raybans
- messenger bag

a post with typograms

2024-04-29T23:36:10+00:00

This is an example post with some typograms code.

```typograms
+----+
|    |---> My first diagram!
+----+
```

Which generates:

+----+
|    |---> My first diagram!
+----+

Another example:

```typograms
.------------------------.
|.----------------------.|
||"https://example.com" ||
|'----------------------'|
| ______________________ |
||                      ||
||   Welcome!           ||
||                      ||
||                      ||
||  .----------------.  ||
||  | username       |  ||
||  '----------------'  ||
||  .----------------.  ||
||  |"*******"       |  ||
||  '----------------'  ||
||                      ||
||  .----------------.  ||
||  |   "Sign-up"    |  ||
||  '----------------'  ||
||                      ||
|+----------------------+|
.------------------------.
```

which generates:

.------------------------.
|.----------------------.|
||"https://example.com" ||
|'----------------------'|
| ______________________ |
||                      ||
||   Welcome!           ||
||                      ||
||                      ||
||  .----------------.  ||
||  | username       |  ||
||  '----------------'  ||
||  .----------------.  ||
||  |"*******"       |  ||
||  '----------------'  ||
||                      ||
||  .----------------.  ||
||  |   "Sign-up"    |  ||
||  '----------------'  ||
||                      ||
|+----------------------+|
.------------------------.

For more examples, check out the typograms documentation.

a post that can be cited

2024-04-28T15:06:00+00:00

This is an example post that can be cited. The content of the post ends here, while the citation information is automatically provided below. The only thing needed is for you to set the citation key in the front matter to true.

a post with pseudo code

2024-04-15T00:01:00+00:00

This is an example post with some pseudo code rendered by pseudocode. The example presented here is the same as the one in the pseudocode.js documentation, with only one simple but important change: everytime you would use $, you should use $$ instead. Also, note that the pseudocode key in the front matter is set to true to enable the rendering of pseudo code. As an example, using this code:

```pseudocode
% This quicksort algorithm is extracted from Chapter 7, Introduction to Algorithms (3rd edition)
\begin{algorithm}
\caption{Quicksort}
\begin{algorithmic}
\PROCEDURE{Quicksort}{$$A, p, r$$}
    \IF{$$p < r$$}
        \STATE $$q = $$ \CALL{Partition}{$$A, p, r$$}
        \STATE \CALL{Quicksort}{$$A, p, q - 1$$}
        \STATE \CALL{Quicksort}{$$A, q + 1, r$$}
    \ENDIF
\ENDPROCEDURE
\PROCEDURE{Partition}{$$A, p, r$$}
    \STATE $$x = A[r]$$
    \STATE $$i = p - 1$$
    \FOR{$$j = p$$ \TO $$r - 1$$}
        \IF{$$A[j] < x$$}
            \STATE $$i = i + 1$$
            \STATE exchange
            $$A[i]$$ with $$A[j]$$
        \ENDIF
        \STATE exchange $$A[i]$$ with $$A[r]$$
    \ENDFOR
\ENDPROCEDURE
\end{algorithmic}
\end{algorithm}
```

Generates:

% This quicksort algorithm is extracted from Chapter 7, Introduction to Algorithms (3rd edition)
\begin{algorithm}
\caption{Quicksort}
\begin{algorithmic}
\PROCEDURE{Quicksort}{$$A, p, r$$}
    \IF{$$p < r$$}
        \STATE $$q = $$ \CALL{Partition}{$$A, p, r$$}
        \STATE \CALL{Quicksort}{$$A, p, q - 1$$}
        \STATE \CALL{Quicksort}{$$A, q + 1, r$$}
    \ENDIF
\ENDPROCEDURE
\PROCEDURE{Partition}{$$A, p, r$$}
    \STATE $$x = A[r]$$
    \STATE $$i = p - 1$$
    \FOR{$$j = p$$ \TO $$r - 1$$}
        \IF{$$A[j] < x$$}
            \STATE $$i = i + 1$$
            \STATE exchange
            $$A[i]$$ with $$A[j]$$
        \ENDIF
        \STATE exchange $$A[i]$$ with $$A[r]$$
    \ENDFOR
\ENDPROCEDURE
\end{algorithmic}
\end{algorithm}

a post with code diff

2024-01-27T19:22:00+00:00

You can display diff code by using the regular markdown syntax:

```diff
diff --git a/sample.js b/sample.js
index 0000001..0ddf2ba
--- a/sample.js
+++ b/sample.js
@@ -1 +1 @@
-console.log("Hello World!")
+console.log("Hello from Diff2Html!")
```

Which generates:

diff --git a/sample.js b/sample.js
index 0000001..0ddf2ba
--- a/sample.js
+++ b/sample.js
@@ -1 +1 @@
-console.log("Hello World!")
+console.log("Hello from Diff2Html!")

But this is difficult to read, specially if you have a large diff. You can use diff2html to display a more readable version of the diff. For this, just use diff2html instead of diff for the code block language:

```diff2html
diff --git a/sample.js b/sample.js
index 0000001..0ddf2ba
--- a/sample.js
+++ b/sample.js
@@ -1 +1 @@
-console.log("Hello World!")
+console.log("Hello from Diff2Html!")
```

If we use a longer example, for example this commit from diff2html, it will generate the following output:

From 2aaae31cc2a37bfff83430c2c914b140bee59b6a Mon Sep 17 00:00:00 2001
From: Rodrigo Fernandes 
Date: Sun, 9 Oct 2016 16:41:54 +0100
Subject: [PATCH 1/2] Initial template override support

---
 scripts/hulk.js                    |  4 ++--
 src/diff2html.js                   |  3 +--
 src/file-list-printer.js           | 11 ++++++++---
 src/hoganjs-utils.js               | 29 +++++++++++++++++------------
 src/html-printer.js                |  6 ++++++
 src/line-by-line-printer.js        |  6 +++++-
 src/side-by-side-printer.js        |  6 +++++-
 test/file-list-printer-tests.js    |  2 +-
 test/hogan-cache-tests.js          | 18 +++++++++++++++---
 test/line-by-line-tests.js         |  3 +--
 test/side-by-side-printer-tests.js |  3 +--
 11 files changed, 62 insertions(+), 29 deletions(-)

diff --git a/scripts/hulk.js b/scripts/hulk.js
index 5a793c18..a4b1a4d5 100755
--- a/scripts/hulk.js
+++ b/scripts/hulk.js
@@ -173,11 +173,11 @@ function namespace(name) {
 // write a template foreach file that matches template extension
 templates = extractFiles(options.argv.remain)
   .map(function(file) {
-    var openedFile = fs.readFileSync(file, 'utf-8');
+    var openedFile = fs.readFileSync(file, 'utf-8').trim();
     var name;
     if (!openedFile) return;
     name = namespace(path.basename(file).replace(/\..*$/, ''));
-    openedFile = removeByteOrderMark(openedFile.trim());
+    openedFile = removeByteOrderMark(openedFile);
     openedFile = wrap(file, name, openedFile);
     if (!options.outputdir) return openedFile;
     fs.writeFileSync(path.join(options.outputdir, name + '.js')
diff --git a/src/diff2html.js b/src/diff2html.js
index 21b0119e..64e138f5 100644
--- a/src/diff2html.js
+++ b/src/diff2html.js
@@ -7,7 +7,6 @@

 (function() {
   var diffParser = require('./diff-parser.js').DiffParser;
-  var fileLister = require('./file-list-printer.js').FileListPrinter;
   var htmlPrinter = require('./html-printer.js').HtmlPrinter;

   function Diff2Html() {
@@ -43,7 +42,7 @@

     var fileList = '';
     if (configOrEmpty.showFiles === true) {
-      fileList = fileLister.generateFileList(diffJson, configOrEmpty);
+      fileList = htmlPrinter.generateFileListSummary(diffJson, configOrEmpty);
     }

     var diffOutput = '';
diff --git a/src/file-list-printer.js b/src/file-list-printer.js
index e408d9b2..1e0a2c61 100644
--- a/src/file-list-printer.js
+++ b/src/file-list-printer.js
@@ -8,11 +8,16 @@
 (function() {
   var printerUtils = require('./printer-utils.js').PrinterUtils;

-  var hoganUtils = require('./hoganjs-utils.js').HoganJsUtils;
+  var hoganUtils;
+
   var baseTemplatesPath = 'file-summary';
   var iconsBaseTemplatesPath = 'icon';

-  function FileListPrinter() {
+  function FileListPrinter(config) {
+    this.config = config;
+
+    var HoganJsUtils = require('./hoganjs-utils.js').HoganJsUtils;
+    hoganUtils = new HoganJsUtils(config);
   }

   FileListPrinter.prototype.generateFileList = function(diffFiles) {
@@ -38,5 +43,5 @@
     });
   };

-  module.exports.FileListPrinter = new FileListPrinter();
+  module.exports.FileListPrinter = FileListPrinter;
 })();
diff --git a/src/hoganjs-utils.js b/src/hoganjs-utils.js
index 9949e5fa..0dda08d7 100644
--- a/src/hoganjs-utils.js
+++ b/src/hoganjs-utils.js
@@ -8,18 +8,19 @@
 (function() {
   var fs = require('fs');
   var path = require('path');
-
   var hogan = require('hogan.js');

   var hoganTemplates = require('./templates/diff2html-templates.js');

-  var templatesPath = path.resolve(__dirname, 'templates');
+  var extraTemplates;

-  function HoganJsUtils() {
+  function HoganJsUtils(configuration) {
+    this.config = configuration || {};
+    extraTemplates = this.config.templates || {};
   }

-  HoganJsUtils.prototype.render = function(namespace, view, params, configuration) {
-    var template = this.template(namespace, view, configuration);
+  HoganJsUtils.prototype.render = function(namespace, view, params) {
+    var template = this.template(namespace, view);
     if (template) {
       return template.render(params);
     }
@@ -27,17 +28,16 @@
     return null;
   };

-  HoganJsUtils.prototype.template = function(namespace, view, configuration) {
-    var config = configuration || {};
+  HoganJsUtils.prototype.template = function(namespace, view) {
     var templateKey = this._templateKey(namespace, view);

-    return this._getTemplate(templateKey, config);
+    return this._getTemplate(templateKey);
   };

-  HoganJsUtils.prototype._getTemplate = function(templateKey, config) {
+  HoganJsUtils.prototype._getTemplate = function(templateKey) {
     var template;

-    if (!config.noCache) {
+    if (!this.config.noCache) {
       template = this._readFromCache(templateKey);
     }

@@ -53,6 +53,7 @@

     try {
       if (fs.readFileSync) {
+        var templatesPath = path.resolve(__dirname, 'templates');
         var templatePath = path.join(templatesPath, templateKey);
         var templateContent = fs.readFileSync(templatePath + '.mustache', 'utf8');
         template = hogan.compile(templateContent);
@@ -66,12 +67,16 @@
   };

   HoganJsUtils.prototype._readFromCache = function(templateKey) {
-    return hoganTemplates[templateKey];
+    return extraTemplates[templateKey] || hoganTemplates[templateKey];
   };

   HoganJsUtils.prototype._templateKey = function(namespace, view) {
     return namespace + '-' + view;
   };

-  module.exports.HoganJsUtils = new HoganJsUtils();
+  HoganJsUtils.prototype.compile = function(templateStr) {
+    return hogan.compile(templateStr);
+  };
+
+  module.exports.HoganJsUtils = HoganJsUtils;
 })();
diff --git a/src/html-printer.js b/src/html-printer.js
index 585d5b66..13f83047 100644
--- a/src/html-printer.js
+++ b/src/html-printer.js
@@ -8,6 +8,7 @@
 (function() {
   var LineByLinePrinter = require('./line-by-line-printer.js').LineByLinePrinter;
   var SideBySidePrinter = require('./side-by-side-printer.js').SideBySidePrinter;
+  var FileListPrinter = require('./file-list-printer.js').FileListPrinter;

   function HtmlPrinter() {
   }
@@ -22,5 +23,10 @@
     return sideBySidePrinter.generateSideBySideJsonHtml(diffFiles);
   };

+  HtmlPrinter.prototype.generateFileListSummary = function(diffJson, config) {
+    var fileListPrinter = new FileListPrinter(config);
+    return fileListPrinter.generateFileList(diffJson);
+  };
+
   module.exports.HtmlPrinter = new HtmlPrinter();
 })();
diff --git a/src/line-by-line-printer.js b/src/line-by-line-printer.js
index b07eb53c..d230bedd 100644
--- a/src/line-by-line-printer.js
+++ b/src/line-by-line-printer.js
@@ -11,7 +11,8 @@
   var utils = require('./utils.js').Utils;
   var Rematch = require('./rematch.js').Rematch;

-  var hoganUtils = require('./hoganjs-utils.js').HoganJsUtils;
+  var hoganUtils;
+
   var genericTemplatesPath = 'generic';
   var baseTemplatesPath = 'line-by-line';
   var iconsBaseTemplatesPath = 'icon';
@@ -19,6 +20,9 @@

   function LineByLinePrinter(config) {
     this.config = config;
+
+    var HoganJsUtils = require('./hoganjs-utils.js').HoganJsUtils;
+    hoganUtils = new HoganJsUtils(config);
   }

   LineByLinePrinter.prototype.makeFileDiffHtml = function(file, diffs) {
diff --git a/src/side-by-side-printer.js b/src/side-by-side-printer.js
index bbf1dc8d..5e3033b3 100644
--- a/src/side-by-side-printer.js
+++ b/src/side-by-side-printer.js
@@ -11,7 +11,8 @@
   var utils = require('./utils.js').Utils;
   var Rematch = require('./rematch.js').Rematch;

-  var hoganUtils = require('./hoganjs-utils.js').HoganJsUtils;
+  var hoganUtils;
+
   var genericTemplatesPath = 'generic';
   var baseTemplatesPath = 'side-by-side';
   var iconsBaseTemplatesPath = 'icon';
@@ -26,6 +27,9 @@

   function SideBySidePrinter(config) {
     this.config = config;
+
+    var HoganJsUtils = require('./hoganjs-utils.js').HoganJsUtils;
+    hoganUtils = new HoganJsUtils(config);
   }

   SideBySidePrinter.prototype.makeDiffHtml = function(file, diffs) {
diff --git a/test/file-list-printer-tests.js b/test/file-list-printer-tests.js
index a502a46f..60ea3208 100644
--- a/test/file-list-printer-tests.js
+++ b/test/file-list-printer-tests.js
@@ -1,6 +1,6 @@
 var assert = require('assert');

-var fileListPrinter = require('../src/file-list-printer.js').FileListPrinter;
+var fileListPrinter = new (require('../src/file-list-printer.js').FileListPrinter)();

 describe('FileListPrinter', function() {
   describe('generateFileList', function() {
diff --git a/test/hogan-cache-tests.js b/test/hogan-cache-tests.js
index 190bf6f8..3bb754ac 100644
--- a/test/hogan-cache-tests.js
+++ b/test/hogan-cache-tests.js
@@ -1,6 +1,6 @@
 var assert = require('assert');

-var HoganJsUtils = require('../src/hoganjs-utils.js').HoganJsUtils;
+var HoganJsUtils = new (require('../src/hoganjs-utils.js').HoganJsUtils)();
 var diffParser = require('../src/diff-parser.js').DiffParser;

 describe('HoganJsUtils', function() {
@@ -21,16 +21,28 @@ describe('HoganJsUtils', function() {
       });
       assert.equal(emptyDiffHtml, result);
     });
+
     it('should render view without cache', function() {
       var result = HoganJsUtils.render('generic', 'empty-diff', {
         contentClass: 'd2h-code-line',
         diffParser: diffParser
       }, {noCache: true});
-      assert.equal(emptyDiffHtml + '\n', result);
+      assert.equal(emptyDiffHtml, result);
     });
+
     it('should return null if template is missing', function() {
-      var result = HoganJsUtils.render('generic', 'missing-template', {}, {noCache: true});
+      var hoganUtils = new (require('../src/hoganjs-utils.js').HoganJsUtils)({noCache: true});
+      var result = hoganUtils.render('generic', 'missing-template', {});
       assert.equal(null, result);
     });
+
+    it('should allow templates to be overridden', function() {
+      var emptyDiffTemplate = HoganJsUtils.compile('');
+
+      var config = {templates: {'generic-empty-diff': emptyDiffTemplate}};
+      var hoganUtils = new (require('../src/hoganjs-utils.js').HoganJsUtils)(config);
+      var result = hoganUtils.render('generic', 'empty-diff', {myName: 'Rodrigo Fernandes'});
+      assert.equal('Rodrigo Fernandes', result);
+    });
   });
 });
diff --git a/test/line-by-line-tests.js b/test/line-by-line-tests.js
index 1cd92073..8869b3df 100644
--- a/test/line-by-line-tests.js
+++ b/test/line-by-line-tests.js
@@ -14,7 +14,7 @@ describe('LineByLinePrinter', function() {
         '            File without changes\n' +
         '        \n' +
         '    \n' +
-        '\n';
+        '';

       assert.equal(expected, fileHtml);
     });
@@ -422,7 +422,6 @@ describe('LineByLinePrinter', function() {
         '        \n' +
         '    \n' +
         '\n' +
-        '\n' +
         '                \n' +
         '            \n' +
         '        \n' +
diff --git a/test/side-by-side-printer-tests.js b/test/side-by-side-printer-tests.js
index 76625f8e..771daaa5 100644
--- a/test/side-by-side-printer-tests.js
+++ b/test/side-by-side-printer-tests.js
@@ -14,7 +14,7 @@ describe('SideBySidePrinter', function() {
         '            File without changes\n' +
         '        \n' +
         '    \n' +
-        '\n';
+        '';

       assert.equal(expectedRight, fileHtml.right);
       assert.equal(expectedLeft, fileHtml.left);
@@ -324,7 +324,6 @@ describe('SideBySidePrinter', function() {
         '        \n' +
         '    \n' +
         '\n' +
-        '\n' +
         '                    \n' +
         '                \n' +
         '            \n' +

From f3cadb96677d0eb82fc2752dc3ffbf35ca9b5bdb Mon Sep 17 00:00:00 2001
From: Rodrigo Fernandes 
Date: Sat, 15 Oct 2016 13:21:22 +0100
Subject: [PATCH 2/2] Allow uncompiled templates

---
 README.md                 |  3 +++
 src/hoganjs-utils.js      |  7 +++++++
 test/hogan-cache-tests.js | 24 +++++++++++++++++++++++-
 3 files changed, 33 insertions(+), 1 deletion(-)

diff --git a/README.md b/README.md
index 132c8a28..46909f25 100644
--- a/README.md
+++ b/README.md
@@ -98,6 +98,9 @@ The HTML output accepts a Javascript object with configuration. Possible options
   - `synchronisedScroll`: scroll both panes in side-by-side mode: `true` or `false`, default is `false`
   - `matchWordsThreshold`: similarity threshold for word matching, default is 0.25
   - `matchingMaxComparisons`: perform at most this much comparisons for line matching a block of changes, default is `2500`
+  - `templates`: object with previously compiled templates to replace parts of the html
+  - `rawTemplates`: object with raw not compiled templates to replace parts of the html
+  > For more information regarding the possible templates look into [src/templates](https://github.com/rtfpessoa/diff2html/tree/master/src/templates)

 ## Diff2HtmlUI Helper

diff --git a/src/hoganjs-utils.js b/src/hoganjs-utils.js
index 0dda08d7..b2e9c275 100644
--- a/src/hoganjs-utils.js
+++ b/src/hoganjs-utils.js
@@ -17,6 +17,13 @@
   function HoganJsUtils(configuration) {
     this.config = configuration || {};
     extraTemplates = this.config.templates || {};
+
+    var rawTemplates = this.config.rawTemplates || {};
+    for (var templateName in rawTemplates) {
+      if (rawTemplates.hasOwnProperty(templateName)) {
+        if (!extraTemplates[templateName]) extraTemplates[templateName] = this.compile(rawTemplates[templateName]);
+      }
+    }
   }

   HoganJsUtils.prototype.render = function(namespace, view, params) {
diff --git a/test/hogan-cache-tests.js b/test/hogan-cache-tests.js
index 3bb754ac..a34839c0 100644
--- a/test/hogan-cache-tests.js
+++ b/test/hogan-cache-tests.js
@@ -36,7 +36,7 @@ describe('HoganJsUtils', function() {
       assert.equal(null, result);
     });

-    it('should allow templates to be overridden', function() {
+    it('should allow templates to be overridden with compiled templates', function() {
       var emptyDiffTemplate = HoganJsUtils.compile('');

       var config = {templates: {'generic-empty-diff': emptyDiffTemplate}};
@@ -44,5 +44,27 @@ describe('HoganJsUtils', function() {
       var result = hoganUtils.render('generic', 'empty-diff', {myName: 'Rodrigo Fernandes'});
       assert.equal('Rodrigo Fernandes', result);
     });
+
+    it('should allow templates to be overridden with uncompiled templates', function() {
+      var emptyDiffTemplate = '';
+
+      var config = {rawTemplates: {'generic-empty-diff': emptyDiffTemplate}};
+      var hoganUtils = new (require('../src/hoganjs-utils.js').HoganJsUtils)(config);
+      var result = hoganUtils.render('generic', 'empty-diff', {myName: 'Rodrigo Fernandes'});
+      assert.equal('Rodrigo Fernandes', result);
+    });
+
+    it('should allow templates to be overridden giving priority to compiled templates', function() {
+      var emptyDiffTemplate = HoganJsUtils.compile('');
+      var emptyDiffTemplateUncompiled = 'Not used!';
+
+      var config = {
+        templates: {'generic-empty-diff': emptyDiffTemplate},
+        rawTemplates: {'generic-empty-diff': emptyDiffTemplateUncompiled}
+      };
+      var hoganUtils = new (require('../src/hoganjs-utils.js').HoganJsUtils)(config);
+      var result = hoganUtils.render('generic', 'empty-diff', {myName: 'Rodrigo Fernandes'});
+      assert.equal('Rodrigo Fernandes', result);
+    });
   });
 });

a post with advanced image components

2024-01-27T11:46:00+00:00

This is an example post with advanced image components.

Image Slider

This is a simple image slider. It uses the Swiper library. Check the examples page for more information of what you can achieve with it.

Image Comparison Slider

This is a simple image comparison slider. It uses the img-comparison-slider library. Check the examples page for more information of what you can achieve with it.

a post with vega lite

2024-01-27T00:20:00+00:00

This is an example post with some vega lite code.

```vega_lite
{
  "$schema": "https://vega.github.io/schema/vega-lite/v5.json",
  "description": "A dot plot showing each movie in the database, and the difference from the average movie rating. The display is sorted by year to visualize everything in sequential order. The graph is for all Movies before 2019.",
  "data": {
    "url": "https://raw.githubusercontent.com/vega/vega/main/docs/data/movies.json"
  },
  "transform": [
    {"filter": "datum['IMDB Rating'] != null"},
    {"filter": {"timeUnit": "year", "field": "Release Date", "range": [null, 2019]}},
    {
      "joinaggregate": [{
        "op": "mean",
        "field": "IMDB Rating",
        "as": "AverageRating"
      }]
    },
    {
      "calculate": "datum['IMDB Rating'] - datum.AverageRating",
      "as": "RatingDelta"
    }
  ],
  "mark": "point",
  "encoding": {
    "x": {
      "field": "Release Date",
      "type": "temporal"
    },
    "y": {
      "field": "RatingDelta",
      "type": "quantitative",
      "title": "Rating Delta"
    },
    "color": {
      "field": "RatingDelta",
      "type": "quantitative",
      "scale": {"domainMid": 0},
      "title": "Rating Delta"
    }
  }
}
```

Which generates:

{
  "$schema": "https://vega.github.io/schema/vega-lite/v5.json",
  "description": "A dot plot showing each movie in the database, and the difference from the average movie rating. The display is sorted by year to visualize everything in sequential order. The graph is for all Movies before 2019.",
  "data": {
    "url": "https://raw.githubusercontent.com/vega/vega/main/docs/data/movies.json"
  },
  "transform": [
    {"filter": "datum['IMDB Rating'] != null"},
    {"filter": {"timeUnit": "year", "field": "Release Date", "range": [null, 2019]}},
    {
      "joinaggregate": [{
        "op": "mean",
        "field": "IMDB Rating",
        "as": "AverageRating"
      }]
    },
    {
      "calculate": "datum['IMDB Rating'] - datum.AverageRating",
      "as": "RatingDelta"
    }
  ],
  "mark": "point",
  "encoding": {
    "x": {
      "field": "Release Date",
      "type": "temporal"
    },
    "y": {
      "field": "RatingDelta",
      "type": "quantitative",
      "title": "Rating Delta"
    },
    "color": {
      "field": "RatingDelta",
      "type": "quantitative",
      "scale": {"domainMid": 0},
      "title": "Rating Delta"
    }
  }
}

This plot supports both light and dark themes.

Cheems908

概率论与数理统计 第4章 随机变量的数字特征

章节概览

4.1 随机变量的期望 (Expectation of a Random Variable)

背景说明

4.1.1 离散型随机变量的期望

4.1.2 连续型随机变量的期望

4.1.3 随机变量函数的期望

4.1.4 多维随机变量函数的期望

4.1.5 期望的性质

4.2 方差 (Variance)

4.2.1 方差的定义

4.2.2 方差的计算公式

4.2.3 方差的性质

4.3 常见分布的数字特征 (Characteristics of Common Distributions)

4.3.1 伯努利分布 (Bernoulli Distribution)

4.3.2 二项分布 (Binomial Distribution)

4.3.3 泊松分布 (Poisson Distribution)

4.3.4 超几何分布 (Hypergeometric Distribution)

4.3.5 负二项分布 (Negative Binomial Distribution)

4.3.6 均匀分布 (Uniform Distribution)

4.3.7 指数分布 (Exponential Distribution)

4.3.8 正态分布 (Normal Distribution)

4.3.9 Gamma分布 (Gamma Distribution)

4.4 切比雪夫不等式 (Chebyshev’s Inequality)

4.4.1 切比雪夫不等式

4.4.2 证明思路

4.5 协方差与相关系数 (Covariance and Correlation)

4.5.1 协方差的定义

4.5.2 协方差与方差的关系

4.5.3 协方差的性质

4.5.4 相关系数的定义

4.5.5 相关系数的性质

4.5.6 相关性的分类

4.5.7 独立性与相关性的关系

4.5.8 二元正态分布的相关性

4.6 矩与协方差矩阵 (Moments and Covariance Matrices)

4.6.1 矩的定义

4.6.2 协方差矩阵的定义

4.6.3 二元正态分布的协方差矩阵表示

4.6.4 非退化情况的概率密度函数

4.6.5 \(n\) 维正态分布

4.6.6 \(n\) 维正态分布的性质

4.6.7 正态随机向量的线性变换

关键术语中英对照表

a post with plotly.js

a post with image galleries

Lightbox2

PhotoSwipe

Spotlight JS

Venobox

a post with tabs

First tabs

Another example

Tabs for something else

a post with typograms

a post that can be cited

a post with pseudo code

a post with code diff

a post with advanced image components

Image Slider

Image Comparison Slider

a post with vega lite

概率论与数理统计第4章随机变量的数字特征