Univariate Normal/Gaussian/Gauss/Laplace-Gauss Distribution/Model/Process (Bell Curve)

Probability Density Function

• $f(X=x)=\frac{1}{𝜎\sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{𝜎}^2}}$
• 𝑓(𝑋=𝑥) = (1/[𝜎*𝑠𝑞𝑟𝑡(2𝜋)])·(𝑒^{-(𝑥-𝜇)²/(2𝜎²)}) # for -∞ < 𝑥 <∞

where:

• 𝜇- mean, location parameter
• 𝜎 - standard deviation, scale/spread parameter

Probability Density Function (Using Precision)

• $f(X=x)=\sqrt{\frac{\beta }{2\pi }}{e}^{-\frac{\beta (x-\mu {)}^2}{2}}$
• 𝑓(𝑋=𝑥) = 𝑠𝑞𝑟𝑡[𝛽/(2𝜋)] 𝑒^{-𝛽(𝑥-𝜇)²/(2)} # for -∞ < 𝑥 <∞

where:

• 𝛽 - precision with interval: (0,∞)

Expectation

• 𝐄[𝑋] = 𝜇

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• 𝐄[𝑋] = _-∞∫^∞𝑥·𝑓(𝑥)·𝑑𝑥 # see this
• 𝐄[𝑋] = _-∞∫^∞ 𝑥·(1/[𝜎*𝑠𝑞𝑟𝑡(2𝜋)])·(𝑒^{-(𝑥-𝜇)²/(2𝜎²)})·𝑑𝑥
• 𝐄[𝑋] = (1/[𝜎*𝑠𝑞𝑟𝑡(2𝜋)])·_-∞∫^∞ 𝑥·(𝑒^{-(𝑥-𝜇)²/(2𝜎²)})·𝑑𝑥
• 𝐄[𝑋] = (1/[𝜎*𝑠𝑞𝑟𝑡(2𝜋)])·_-∞∫^∞ 𝑥·(𝑒^{-(𝑥-𝜇)²/(2𝜎²)})·𝑑𝑥

• 𝐄[𝑋] = TODO # integration by parts

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• ∫𝑥·(𝑒^{-(𝑥-𝜇)²/(2𝜎²)})·𝑑𝑥
• ∫𝑥·(𝑒^{-(𝑥-𝜇)²/(2𝜎²)})·𝑑𝑥
• ∫𝑢·𝑑𝑣 = 𝑢·𝑣 - ∫𝑣·𝑑𝑢

TODO

- <font style="color: rgb(255,0,255);">𝑢 = 𝑥</font>
- <font style="color: rgb(255,102,0);">𝑑𝑣 = (𝑒<sup>-(𝑥-𝜇)²/(2𝜎²)</sup>)·𝑑𝑥</font>
- <font style="color: rgb(0,128,0);">𝑑𝑢 = 1</font>
- <font style="color: rgb(51,204,204);">𝑣 = -\[𝜎²/(𝑥-𝜇)\]·(𝑒<sup>-(𝑥-𝜇)²/(2𝜎²)</sup>)</font>

• ∫𝑥·(𝑒^{-(𝑥-𝜇)²/(2𝜎²)})·𝑑𝑥 = 𝑥·-[𝜎²/(𝑥-𝜇)]·(𝑒^{-(𝑥-𝜇)²/(2𝜎²)}) - ∫-[𝜎²/(𝑥-𝜇)]·(𝑒^{-(𝑥-𝜇)²/(2𝜎²)})·1
• ∫𝑥·(𝑒^{-(𝑥-𝜇)²/(2𝜎²)})·𝑑𝑥 = -𝑥·[𝜎²/(𝑥-𝜇)]·(𝑒^{-(𝑥-𝜇)²/(2𝜎²)}) + ∫[𝜎²/(𝑥-𝜇)]·(𝑒^{-(𝑥-𝜇)²/(2𝜎²)})·1

• • 𝑢 = (𝑒^{-(𝑥-𝜇)²/(2𝜎²)})
  • 𝑑𝑣 = 𝑥·𝑑𝑥
  • 𝑑𝑢 = -(1/𝜎²)·(𝑥-𝜇)·(𝑒^{-(𝑥-𝜇)²/(2𝜎²)})
  • 𝑣 = (1/2)𝑥²
• ∫𝑥·(𝑒^{-(𝑥-𝜇)²/(2𝜎²)})·𝑑𝑥 = (𝑒^{-(𝑥-𝜇)²/(2𝜎²)})·(1/2)𝑥² - ∫(1/2)𝑥²·-(1/𝜎²)·(𝑥-𝜇)·(𝑒^{-(𝑥-𝜇)²/(2𝜎²)})

• TODO

Variance

𝑉𝑎𝑟(𝑋) = 𝜎²

Subpages

• z-distribution (Standard Normal Distribution)
• Univariate Normal Distribution - Algebra
• Univariate Normal Distribution - Derivation

Regression / Learning 𝜇 & 𝜎 Parameters

• see Gaussian Process Regression (GPR) - Kriging