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正态分布 N(μ, σ²)
Normal distribution N(μ, σ²)
钟形曲线 · μ 决定中心,σ 决定胖瘦
Bell curve · μ centres it, σ sets the spread
二项分布
Binomial distribution
P(X=k) = C(n,k)·pᵏ·(1−p)ⁿ⁻ᵏ · n 次试验中成功 k 次
P(X=k) = C(n,k)·pᵏ·(1−p)ⁿ⁻ᵏ · k successes in n trials
单样本 z 检验
One-sample z-test
z=(x̄−μ0)/(σ/√n) · p 值/临界值 · 单/双尾拒绝域
z = (x̄ − μ₀) / (σ/√n) · p-value from the normal CDF · shaded rejection region
均值的置信区间
Confidence interval for a mean
x̄ ± z*·σ/√n · 置信水平/边际误差 · 覆盖率
CI = x̄ ± z*·(σ/√n) · width shrinks with larger n, grows with higher confidence
Student t 分布 vs 正态
Student's t-distribution vs normal
t vs 正态 · 自由度/重尾 · 临界 t*
Heavier tails at low df, converging to N(0,1) as df → ∞ · critical t* vs z*
最小二乘线性回归
Ordinary least-squares regression
最小二乘 b,a · 拟合线 + 残差 · R²
Fit a line by minimising the squared residuals · R² and correlation r
皮尔逊相关系数
Pearson correlation coefficient
皮尔逊 r=cov/(σxσy) · 散点 · 线性相关强度
How tightly two variables move together along a line · r ∈ [−1, +1]
逻辑回归(sigmoid 二分类)
Logistic regression (sigmoid classification)
sigmoid p=1/(1+e^−(β0+β1x)) · 决策边界 · 准确率
A linear score β0 + β1·x squashed into a probability, then thresholded
中心极限定理
Central Limit Theorem
母体 → 样本均值分布 · SE=σ/√n · 趋于正态
Sampling distribution of the mean narrows to σ/√n and turns normal — whatever the parent
泊松分布
Poisson distribution
PMF λ^k·e^−λ/k! · 均值=方差=λ · 稀有事件
PMF of rare events · mean = variance = λ · normal approximation overlaid
二维主成分分析(PCA)
Principal Component Analysis (2D)
协方差特征向量 · 主轴 · PC1 方差解释比
Covariance → eigenvectors → the axis of maximum variance
卡方检验 · 拟合优度与独立性
Chi-square test · goodness-of-fit & independence
χ²=Σ(O−E)²/E · 自由度/p 值 · 观测 vs 期望
Edit the counts; expected values, χ², degrees of freedom and the p-value recompute exactly
单因素方差分析 · F 检验
One-way ANOVA · F-test
组间/组内 SS · F=MSB/MSW · 拒绝域
Adjust the group means, spread and sizes; SSB, SSW and F recompute exactly
贝叶斯更新 · Beta-二项共轭
Bayesian updating · Beta-Binomial conjugate
Beta(α,β) → Beta(α+k,β+n−k) · 后验均值 · 可信区间
A Beta prior plus binomial data gives a Beta posterior in closed form
The birthday paradox
The birthday paradox
P=1−365!/((365−n)!·365^n) · n=23 约50% · 曲线
P(at least two share a birthday) = 1 − (365·364·…·(365−n+1)) / 365ⁿ
The Monty Hall problem
The Monty Hall problem
三门一车 · 主持人开败门 · 坚持1/3 vs 换门2/3 · 全枚举
Pick a door, the host reveals a goat — should you switch? Staying wins 1/3, switching wins 2/3.
Bertrand's box paradox
Bertrand's box paradox
GG/GS/SS 三盒 · 抽到金则另一枚也金=2/3 · 案例计数
You drew a gold coin — is the other coin in the box gold? The answer is 2/3, not 1/2.
Buffon's needle
Buffon's needle
跨线概率 2L/(πd) · 确定性针阵 · 估计 π
P(cross) = 2L / (πd) · a needle-drop estimate of π
Galton board (quincunx)
Galton board (quincunx)
n排钉→二项 C(n,k)/2^n · 逼近正态 · 精确桶计数
Bin k gets C(n,k)/2ⁿ · the binomial B(n, ½) → normal
1-D random walk
1-D random walk
±1步 · E[x]=0 · Var=n · 典型距离√n · 末位二项
±1 steps · E[X] = 0, Var = n, typical distance ≈ √n
Gambler's ruin · absorbing random walk
Gambler's ruin · absorbing random walk
破产概率(1−(q/p)^i)/(1−(q/p)^N) · 公平(N−i)/N
Bet $1 a round until you hit $0 or $N — the ruin probability is closed-form
Coupon collector · harmonic-number expectation
Coupon collector · harmonic-number expectation
集齐n种期望 n·H_n=n·Σ1/k · 逐券等待
Collect all n coupons — the expected number of draws is exactly n·Hₙ
Secretary problem · optimal stopping
Secretary problem · optimal stopping
最优停止 · 拒前 n/e≈37% · 成功概率→1/e
Look then leap — reject the first r, take the next record-breaker
Expected value & variance
Expected value & variance
E[X]=Σx·p · Var=Σ(x−μ)²p · SD=√Var · 可调分布
E[X] = Σ x·p(x) · Var[X] = Σ (x−μ)²·p(x) = E[X²] − (E[X])² · SD = √Var
The inclusion–exclusion principle
The inclusion–exclusion principle
|A∪B∪C|=ΣA−Σ(A∩B)+(A∩B∩C) · 维恩 · 加减加
|A∪B∪C| = |A| + |B| + |C| − |A∩B| − |A∩C| − |B∩C| + |A∩B∩C|
The St. Petersburg paradox
The St. Petersburg paradox
首正面第n次赔2^n · E=Σ(1/2^n·2^n)=∞ · 有限本金→有限公平价
Pay $2ⁿ if the first heads is on flip n · E[payoff] = Σ (1/2ⁿ)·2ⁿ = 1 + 1 + 1 + … = ∞