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

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

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

Probability Density Function

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  • 𝑓(𝑋=π‘₯) = (1/[𝜎*π‘ π‘žπ‘Ÿπ‘‘(2πœ‹)])·(𝑒-(π‘₯-πœ‡/(2𝜎²)) # for -∞ < π‘₯ <∞

where:

Probability Density Function (Using Precision)

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  • 𝑓(𝑋=π‘₯) = π‘ π‘žπ‘Ÿπ‘‘[𝛽/(2πœ‹)] 𝑒-𝛽(π‘₯-πœ‡)²/(2) # for -∞ < π‘₯ <∞

where:

  • 𝛽 - precision with interval: (0,∞)

Expectation

  • 𝐄[𝑋] = πœ‡
 Click here to expand...
  • 𝐄[𝑋] = -∞π‘₯·π‘“(π‘₯)·π‘‘π‘₯ # see this
  • 𝐄[𝑋] = -∞ π‘₯·(1/[𝜎*π‘ π‘žπ‘Ÿπ‘‘(2πœ‹)])·(𝑒-(π‘₯-πœ‡)²/(2𝜎²))·π‘‘π‘₯
  • 𝐄[𝑋] = (1/[𝜎*π‘ π‘žπ‘Ÿπ‘‘(2πœ‹)])·-∞ π‘₯·(𝑒-(π‘₯-πœ‡)²/(2𝜎²))·π‘‘π‘₯
  • 𝐄[𝑋] = (1/[𝜎*π‘ π‘žπ‘Ÿπ‘‘(2πœ‹)])·-∞ π‘₯·(𝑒-(π‘₯-πœ‡)²/(2𝜎²))·π‘‘π‘₯
  • 𝐄[𝑋] = TODO # integration by parts
    •  Click here to expand...
      • π‘₯·(𝑒-(π‘₯-πœ‡)²/(2𝜎²))·π‘‘π‘₯
      • π‘₯·(𝑒-(π‘₯-πœ‡)²/(2𝜎²))·π‘‘π‘₯
      • 𝑒·π‘‘𝑣𝑒·π‘£ - ∫𝑣·π‘‘𝑒

      TODO

        • 𝑒 = π‘₯
        • 𝑑𝑣 = (𝑒-(π‘₯-πœ‡)²/(2𝜎²))·π‘‘π‘₯
        • 𝑑𝑒 = 1
        • 𝑣 = -[𝜎²/(π‘₯-πœ‡)]·(𝑒-(π‘₯-πœ‡)²/(2𝜎²))
      • π‘₯·(𝑒-(π‘₯-πœ‡)²/(2𝜎²))·π‘‘π‘₯ = π‘₯·-[𝜎²/(π‘₯-πœ‡)]·(𝑒-(π‘₯-πœ‡)²/(2𝜎²)) - ∫-[𝜎²/(π‘₯-πœ‡)]·(𝑒-(π‘₯-πœ‡)²/(2𝜎²))·1
      • π‘₯·(𝑒-(π‘₯-πœ‡)²/(2𝜎²))·π‘‘π‘₯ = -π‘₯·[𝜎2/(π‘₯-πœ‡)]·(𝑒-(π‘₯-πœ‡)²/(2𝜎²)) + ∫[𝜎²/(π‘₯-πœ‡)]·(𝑒-(π‘₯-πœ‡)²/(2𝜎²))·1


        • 𝑒 = (𝑒-(π‘₯-πœ‡)²/(2𝜎²))
        • 𝑑𝑣 = π‘₯·π‘‘π‘₯
        • 𝑑𝑒 = -(1/𝜎²)·(π‘₯-πœ‡)·(𝑒-(π‘₯-πœ‡)²/(2𝜎²))
        • 𝑣 = (1/2)π‘₯²
      • π‘₯·(𝑒-(π‘₯-πœ‡)²/(2𝜎²))·π‘‘π‘₯ = (𝑒-(π‘₯-πœ‡)²/(2𝜎²))·(1/2)π‘₯² - ∫(1/2)π‘₯²·-(1/𝜎²)·(π‘₯-πœ‡)·(𝑒-(π‘₯-πœ‡)²/(2𝜎²))
  • TODO

Variance

π‘‰π‘Žπ‘Ÿ(𝑋) = 𝜎²


Regression / Learning πœ‡ & 𝜎 Parameters