concepts

statitics

box whisker

![[_resources/Pasted image 20260320093710.png]]

left skew/right skew: 1. 哪里腿长(first compare whisker, then compareMed-Q1 or Q3-Med) 2. =哪里skew; above is left skew

\[ Y = aX + b \quad \Rightarrow \quad \mathrm{Var}(Y) = a^2 \mathrm{Var}(X) \]

Probability

if \(A\) and \(B\) are independent, \(P(A) \times P(B) = P(A \cap B)\)

Distribution

Normal Distribution

empirical rule: 68, 95, 99.7

![[_resources/Pasted image 20260409101750.png]]

敌人图鉴

permutation and combination

分堆问题：

9 items divide into two groups of 4 and 5: 9C4

8 items divide into two equal group: 8C4 / 2!

9 items divide into 3 equal groups: 9C3 * 6C3 / 3!

9 items divide into 3 groups: 2, 2, 5: 9C2 * 7C2 / 2!

捆绑法

插9 items divide into 3 groups: 2, 2, 5: 9C2 * 9C2 / 2! is this correct?板法

正难则反：total cases - complementary cases

probability

\(\Sigma{P(x)}=1\) and \(P(x)\) always has range \([0,1]\)

draw cards without replacement, 抽一张分子和分母减一张

probability density function: - ask for probability over interval: integration

Statistics GDC

one-variable (with frequency) 1. spreadsheet 2. frequency(f) and x as variable names of columns 3. enter data 4. menu -> statistics -> stat calculation -> one variable 5. x1 list: x; frequency list: f

two-variable: - \(r>0\) positive relationship; \(r<0\) negative relationship - \(y=a_1 x+b_1\) can only used to predict \(y\); \(x=a_2 y+b_2\) vice versa; both pass through \((\bar{x},\bar{y})\)

normal distribution

z-score: transform a given normal distribution to the standard form, so that \(P(X>x_\text{original value}) = P(Z > z_\text{position in standard form})\)

已知分布和标准分布，求约束条件下的概率：用标准分布公式表示约束，注意\(X~N(\mu, \sigma^2)\)第二个参数是方差而非标准差

例子： ![[_resources/Pasted image 20260215151203.png]]

GDC

normal CDF: get possibility(area) from boundary

inverse normal CDF: get right boundary from possibility(area), and assume left boundary is \(- \infty\)

Binomial Distribution

GDC

binomial PDF: get probability of exact \(n\) successes from \(k\) total trials and success probability

binomial CDF: get cumulative probability of at most \(n\) success from \(k\) total trials and success probability

reversed binomial: get the \(n\) successes from cumulative success probability, total trial number, single success probability

reversed binomial N: get the number of total trials from success number and success probability that is larger than the cumulative probability