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