Verification of integrals

Gaussian definition

p[x_] := PDF[NormalDistribution[μ, σ], x] ;

p[x] // TraditionalForm

^(-(x - μ)^2/(2 σ^2))/((2 π)^(1/2) σ)

Integrate to 1?

∫_ (-∞)^∞ p[x] x

If[Re[σ^2] >0, (2 π)^(1/2) σ^2^(1/2), ∫_ (-∞)^∞^(-(x - μ)^2/(2 σ^2)) x]/((2 π)^(1/2) σ)

Mean μ

 ∫_ (-∞)^∞x p[x] x

If[Re[σ^2] >0, (2 π)^(1/2) μ σ^2^(1/2), ∫_ (-∞)^∞^(-(x - μ)^2/(2 σ^2)) xx]/((2 π)^(1/2) σ)

Variance σ^2

 ∫_ (-∞)^∞ (x - μ)^2 p[x] x

If[Re[σ^2] >0, ((2 π)^(1/2) σ^4)/σ^2^(1/2), ∫_ (-∞)^∞^(-(x - μ)^2/(2 σ^2)) (x - μ)^2x]/((2 π)^(1/2) σ)


Created by Mathematica  (September 8, 2003)