Central Limit Theorem (CLT) 中央極限定理

When
population mean = μ
population variance = σ²

random sample of size n
random samples: X₁, X₂, ..., X_n,

z-score/z-value/standard score z分數/標準化值 (Wikipedia/維基百科)

sum of samples/sample sum 樣本和 S_n = (X₁ + X₂ + ... + X_n) = ΣX_i where i = 1 to n
sample mean 樣本平均數 x̄ = (X₁ + X₂ + ... + X_n)/n = S_n/n
sample variance 樣本變異數 s²
sample standard deviation 樣本標準差 s
sample size n

if σ is known:

Z = (x̄ - μ)/(σ/√ n) = (Sn - n)/σ√n

When n → ∞ (n >= 30),

Z ~ N(0, 1)

One special case of gamma distribution:
chi-square distribution 卡方分布 (χ²) α = v/2 or n/2, β = 2 for gamma distribution.

χ² = (n - 1)s²/σ² with n-1 degrees of freedom.

if σ is unknown and n is large:

s ≈ σ

Student t-distribution/Student's t-distribution 學生t分布
t-distribution t分布

T = (x̄ - μ)/(s/√ n)  with n-1 degrees of freedom.

t-table:
row: probability α at t_α
column - degree of freedom

參考資料

Essentials of Probability & Statistics for Engineers & Scientists (Walpole at el.)，機率與統計 (繆紹綱譯，滄海2013)，p201, 242-248

機率與統計 (陳鍾誠網站)

Mathematical statistics with applications in R, Ramachandran & Tsokos, 2nd edition (2015), p162-164, 191-194

Student's t-distribution (Wikipedia)

Normal Distribution 常態分布/常態分配

normal distribution 常態分布/常態分配
Gaussian distribution 高斯分布

bell-shaped 鐘形

If random variable X has a normal distribution N(μ,σ²),
the probability density function (pdf) of X is:

f(x) = [1/√(2πσ²)]exp[-(x-μ)²/2σ²]

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Excel formulas:

NORM.DIST(X, μ, σ, cumulative)

if cumulative = true => cdf
if cumulative = false => pdf

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Z-score/Z-value/Z-transform/standard score z分數/z轉換/標準化值 (Wikipedia/維基百科)
How to get the Z-tansform from normal random variable X:
Z = (X - μ)/σ where μ = population mean and σ = populaiton variance

Z 是N(0,1) 的標準常態分布

(Note: The Z-transform used in statistics is not the Z-transform used is digital signal processing. 此處的Z轉換與數位訊號處理DSP中的Z轉換不同)

standard normal probability distribution 標準常態機率分布 N(0,1)
mean μ = 0 and variance σ² = 1

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Excel Formula:

Z = STANDARDIZE(X, μ, σ)

Inverse Z ie get Z score with known cumulative probability (area under curve):
e.g. α=5% => Z_α/2 = NORM.INV(1-α/2, μ, σ) = NORM.INV(1-2.5%, 0, 1) since μ = 0 and σ = 1 for the standard normal distribution.

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If X1...Xn are normally distributed as N(μ,σ²),
ΣZ_i² = Σ((X - μ)/σ)² is a χ² distribution with n degrees of freedom.
ΣX_i² is a χ² distribution with n degrees of freedom.

參考資料

Essentials of Probability & Statistics for Engineers & Scientists (Walpole at el.)，機率與統計 (繆紹綱譯，滄海2013)

Mathematical statistics with applications in R, Ramachandran & Tsokos, 2nd edition (2015), p188

Sampling Schemes 抽樣方法

抽樣方法可分為：
隨機抽樣
非隨機抽樣

隨機抽樣
simple random sampling 簡單隨機抽樣 - every element may be selected with equal chance
systematic sampling 系統抽樣/系統性抽樣 - every Kth element, e.g, 3, 10, 17, 24, ... (K=7)
stratified sampling 分層抽樣 - e.g. student age, university, major
cluster sampling 部落抽樣/叢集抽樣
multiphase sampling 多相抽樣 multistage sampling 多段抽樣

非隨機抽樣
convenience sampling 偶遇抽樣/方便抽樣
quota sampling 配額抽樣/定額抽樣
purposive sampling 主觀抽樣 / judgmental sampling 立意抽樣
snowball sampling 滾雪球抽樣

validity 效度

參考資料

Essentials of Probability & Statistics for Engineers & Scientists (Walpole at el.)，機率與統計 (繆紹綱譯，滄海2013), p8

Mathematical statistics with applications in R, Ramachandran & Tsokos, 2nd edition (2015), p8-12

統計學-李柏堅-第01章：抽樣 (CUSTCourses影片)

Mean, Variance and Standard Deviation 平均數、變異數、標準差

mean 平均數
standard deviation 標準差

population mean 母體平均數 μ = Σx_i /N = (x₁+x₂+...+x_N)/N, where i = 1 to N
population variance 母體變異數 σ² = Σ(x_i - μ)²/N, where i = 1 to N
population standard deviation 母體標準差 σ

sample mean 樣本平均數 x̄ = Σx_i /n = (x₁+x₂+...+x_n)/n, where i = 1 to n
sample variance 樣本變異數 s² = Σ(x_i - x̄)²/(n - 1), where i = 1 to n
sample standard deviation 樣本標準差 s

Note:
求標準差時只要將變異數開根號即可
σ² 和 s²的分母不同，母體變異數 σ² 的分母是N(或n)，樣本變異數 s²的分母則是n - 1

Excel Formulas:

mean - AVERAGE()

population variance - VAR.P()
sample variance - VAR.S()

population standard deviation - STDEV.P()
sample standard deviation - STDEV.S()

參考資料

Population Mean and Sample Mean

Sample Standard Deviation vs. Population Standard Deviation

統計學-李柏堅-第01章：標準差 (CUSTCourses影片)

Mathematical statistics with applications in R, Ramachandran & Tsokos, 2nd edition (2015), p26-27

Eye Anatomy 眼睛構造名詞

眼睛有三層薄膜(membranes)：
1. cornea & sclera 角膜&鞏膜
2. uvea 葡萄膜 (choroid 脈絡膜 + iris 虹膜 + ciliary body 睫狀體)
3. retina 視網膜

lens 水晶體/晶狀體
pupil 瞳孔
iris 虹膜
cornea 角膜
zonular fibers 懸韌帶
ciliary body 睫狀體
aqueous humor 房水/水狀液
anterior segment/anterior humor/anterior chamber 前段/前房
vitreous humor/vitreous chamber 玻璃體
retina 視網膜
choroid 脈絡膜
sclera 鞏膜
fovea 窩/凹 - with cone cells only
blood vessels 血管
optic nerve 視神經
optic disk 視神經盤 blind spot 盲點
macula lutea 黃斑

ganglion cell 神經節細胞
amacrine cell 無長突細胞/無軸突細胞
bipolar cell 雙極細胞
horizontal cell 水平細胞
retinal pigment epithelium 視網膜色素上皮

photoreceptor 感光體/光受器
rod cell 桿狀細胞 - sensitive to light level
cone cell 錐狀細胞 - sensitive to color

orbit 眼眶
eye extrinsic muscles 眼外肌
oculomotor muscle 眼動肌
rectus 直肌
oblique 斜
superior oblique muscle 上斜肌
superior rectus muscle 上直肌
inferior oblique muscle 下斜肌
inferior rectus muscle 下直肌
medial rectus muscle 內直肌
lateral rectus muscle 外直肌
electrooculogram/electrooculography (EOG) 眼電圖/眼動圖

相關資料

Human Eye (Wikipedia)
眼外在肌(Eye extrinsic muscles) (小小整理站)
聽覺原理與聽損原因 (電子耳資訊小站)

Moment-Generating Function 動差生成函數/力矩產生函數

M_x(t) = E(e^tx) = Expected value of exponential function exp(tx).

M_x(t) = 1 + E[1 + tX + (tX)²/2! + ... + (tX)ⁿ/n!

Derivatives of the moment-generation function with t=0:

M_x⁽ⁿ⁾(0) = E[Xⁿ],         n = 1, 2, 3, ...

first moment - mean 平均數

second moment - variance 變異數

third moment - skewness 偏度/歪度/偏態

fourth moment - kurtosis 尖度/峰度

參考資料

動差生成函數 (Moment Generation Function) (陳鍾誠網站)

Mathematical statistics with applications in R, Ramachandran & Tsokos, 2nd edition (2015),  p96-100 (Section 2.6)