Gauss Probability Workshop

STUDENT #1 NAME:                                      STUDENT #2 NAME:

The Gauss probability distribution has the form

W[n]=A (e^(-(((((n - μ))^2))/(2(σ^2)))))

Problem #1:  Determine the constant  A using the normalization condition

(∫_0^∞) W[n] dn  =  
      1

Hint:  Extend the integral to -∞ in order to simplify the integration.  This is permitted assuming W[n] is zero for negative n. Your answer for the Gauss distribution should be

W[n]=(1/(Sqrt[2 π (σ^2)])) (e^(-(((((n - μ))^2))/(2 (σ^2)))))

Problem #2:  Calculate the average of the Gauss distribution

\(\(\(\ \)\(n\&_ = ∫\_0\%∞ 
          n\ W[n] \(\(dn\)\(\ \)\)\)\)\)

Hint:  Again extend the integral to -∞ in order to simplify the integration.

Problem #3:  Calculate the standard deviation of the Gauss distribution using

\(\(\((n - μ)\)\^2\)\&_ = \ \ ∫\_0\%∞\((n - μ)\)\^2\ W[n] \(\(dn\)\(\ \)\)\)

Problem #4:  Calculate the error integral

(∫_(μ - σ)^(μ + σ))W[
              n] dn=(1/(Sqrt[2 π (σ^2)]))(∫_(μ - σ)^(μ + σ))(e^(-(((((n - μ))^2))/(2 (σ^2)))))\(\(dn\)\(\ \)\)

and give the numerical value of the integral.

Problem #5:  Calculate the error integral

\(∫\_\(μ - 2  σ\)\%\(μ + 2  σ\)W[n] dn\)=(1/(Sqrt[2 π (σ^2)]))(∫_(μ - 2  σ)^(μ + 
                  2  σ))(e^(-(((((n - μ))^2))/(2 (σ^2)))))\(\(dn\)\(\ \)\)

and give the numerical value of the integral.

Problem #6:  Graph using Plot the Gauss distribution

W[n]=(1/(Sqrt[2 π (σ^2)])) (e^(-(((((n - μ))^2))/(2 (σ^2)))))

using μ=8 and σ=2.  Where is the location of the peak of the distribution?

Problem #7:  Graph using Plot the Gauss distribution

W[n]=(1/(Sqrt[2 π (σ^2)])) (e^(-(((((n - μ))^2))/(2 (σ^2)))))

using μ=8 and σ=3.  In comparison with problem #6, how has the width of the distribution changed?