proof that normal distribution is a distribution

-e-(x-μ)22σ2σ2π𝑑x = (-e-(x-μ)22σ2σ2π𝑑x)2
= -e-(x-μ)22σ2σ2π𝑑x-e-(y-μ)22σ2σ2π𝑑y
= --e-(x-μ)2+(y-μ)22σ2σ22π𝑑x𝑑y

Substitute x=x-μ and y=y-μ. Since the bounds are infinite, they do not change, and dx=dx and dy=dy. Thus, we have

--e-(x-μ)2+(y-μ)22σ2σ22π𝑑x𝑑y = --e-(x)2+(y)22σ2σ22π𝑑x𝑑y.

Converting to polar coordinates, we obtain

--e-(x)2+(y)22σ2σ22π𝑑x𝑑y = 002πre-r22σ2σ22π𝑑r𝑑θ
= 02πdθ2π0re-r22σ2σ2𝑑r
= θ2π|02π1σ20re-r22σ2𝑑r
= 2π2πσ2σ2(-e-r22σ2)|0
= 11
= 1.
Title proof that normal distributionMathworldPlanetmath is a distributionPlanetmathPlanetmath
Canonical name ProofThatNormalDistributionIsADistribution
Date of creation 2013-03-22 13:29:12
Last modified on 2013-03-22 13:29:12
Owner Wkbj79 (1863)
Last modified by Wkbj79 (1863)
Numerical id 7
Author Wkbj79 (1863)
Entry type Proof
Classification msc 62E15
Related topic AreaUnderGaussianCurve