Apa itu Integral Gaussian?

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Bell curve e^(−x²): area = √π

The integral of e^(−x²) over all x equals √π ≈ 1.7725. This is the Gaussian integral. Its square root divided by √(2π) gives the standard normal distribution curve.

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Normal distribution formula

f(x) = (1/σ√(2π)) · e^(−(x−μ)²/2σ²)

σ = stdanard deviation, μ = mean

The 1/√(2π) normalisation factor comes directly from the Gaussian integral: ∫e^(−x²)dx = √π.

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The squaring trick: ∫e^(−x²)dx = √π

I² = ∫∫ e^(−x²−y²) dx dy = ∫₀^∞ e^(−r²) 2πr dr = π

Step 1: Square I – convert to double integral over bidang

Step 2: Switch to polar coordinates (r, θ) – the θ integral gives 2π

Step 3: Substitute u = r² – the r integral gives 1/2. Therefore I² = π, so I = √π.

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Bagaimana hubungan integral Gauss dengan distribusi normal?

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