The Error Function and Normal Distribution: Norm(x) and Erf(x)
Introduction
The error function is defined as:
erf(x) = 2/√π * ∫( e^(-t^2) dt, 0, x)
norm(x) = 1/√(2*π) * ∫( e^(-z^2/2) dx, -∞, x)
We were assuming that the mean μ = 0 and variance σ = 1.
Derivation: norm(x) in terms of erf(x)
norm(x)
= 1/√(2*π) * ∫( e^(-z^2/2) dx, -∞, x)
= 1/√(2*π) * ∫( e^(-z^2/2) dx, -∞, 0) + 1/√(2*π) * ∫( e^(-z^2/2) dx, 0, x)
= 1/2 + 1/√(2*π) * ∫( e^(-z^2/2) dx, 0, x)
-----------------------------
Substitution:
t^2 = z^2/2
t = z/√2
√2 * t = z
√2 dt = dz
z = 0, t = 0
z = x, t = x/√2
-----------------------------
= 1/2 + 1/√(2*π) * ∫( e^(-z^2/2) dx, 0, x)
= 1/2 + 1/√(2*π) * ∫(√2 * e^(-t^2) dt, 0, x/√2)
= 1/2 + 1/√π * √π/2 * 2/√π * ∫(√2 * e^(-t^2) dt, 0, x/√2)
= 1/2 + (1/√π * √π/2) * (2/√π * ∫(√2 * e^(-t^2) dt, 0, x/√2))
= 1/2 + 1/2 * 2/√π * ∫(√2 * e^(-t^2) dt, 0, x/√2)
= 1/2 + 1/2 *erf(x/√2)
norm(x) = 1/2 + 1/2 *erf(x/√2)
norm(x) - 1/2 = 1/2 *erf(x/√2)
2 norm(x) - 1 = erf(x/√2)
Let t = x/√2, then:
2 norm(√2 * t) - 1 = erf(t)
Summary
norm(x) = 1/2 + 1/2 * erf(x/√2)
erf(t) = 2 * norm(√2 * t) - 1
with μ = 0 and σ = 1.
Examples
norm(1) = 1/2 + 1/2 * erf(1/√2) ≈ 0.841344746069
erf(1) = 2 * norm(√2) - 1 ≈ 0.84270079295
Eddie
All original content copyright, © 2011-2019. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
Introduction
The error function is defined as:
erf(x) = 2/√π * ∫( e^(-t^2) dt, 0, x)
norm(x) = 1/√(2*π) * ∫( e^(-z^2/2) dx, -∞, x)
We were assuming that the mean μ = 0 and variance σ = 1.
Derivation: norm(x) in terms of erf(x)
norm(x)
= 1/√(2*π) * ∫( e^(-z^2/2) dx, -∞, x)
= 1/√(2*π) * ∫( e^(-z^2/2) dx, -∞, 0) + 1/√(2*π) * ∫( e^(-z^2/2) dx, 0, x)
= 1/2 + 1/√(2*π) * ∫( e^(-z^2/2) dx, 0, x)
-----------------------------
Substitution:
t^2 = z^2/2
t = z/√2
√2 * t = z
√2 dt = dz
z = 0, t = 0
z = x, t = x/√2
-----------------------------
= 1/2 + 1/√(2*π) * ∫( e^(-z^2/2) dx, 0, x)
= 1/2 + 1/√(2*π) * ∫(√2 * e^(-t^2) dt, 0, x/√2)
= 1/2 + 1/√π * √π/2 * 2/√π * ∫(√2 * e^(-t^2) dt, 0, x/√2)
= 1/2 + (1/√π * √π/2) * (2/√π * ∫(√2 * e^(-t^2) dt, 0, x/√2))
= 1/2 + 1/2 * 2/√π * ∫(√2 * e^(-t^2) dt, 0, x/√2)
= 1/2 + 1/2 *erf(x/√2)
norm(x) = 1/2 + 1/2 *erf(x/√2)
norm(x) - 1/2 = 1/2 *erf(x/√2)
2 norm(x) - 1 = erf(x/√2)
Let t = x/√2, then:
2 norm(√2 * t) - 1 = erf(t)
Summary
norm(x) = 1/2 + 1/2 * erf(x/√2)
erf(t) = 2 * norm(√2 * t) - 1
with μ = 0 and σ = 1.
Examples
norm(1) = 1/2 + 1/2 * erf(1/√2) ≈ 0.841344746069
erf(1) = 2 * norm(√2) - 1 ≈ 0.84270079295
Eddie
All original content copyright, © 2011-2019. Edward Shore. Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited. This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.
