Smoothing the distribution
 Approximation of N_{g}(μ, σ^{2}):
 discrete distribution cumbersome for large values (≙
high precision)
 idea
 approximate N_{g} with a continuous
distribution N_{s} ("smoothed" normal)
 interprete rounding as a special noise
 concrete model
 X_{g} = X + Y
 where
 X ∼ N(μ, σ^{2})
 X_{g} = round(X)
 Y interpreted as noise with
 Y ∼ U(−0.5, 0.5)
 useful approach in computer arithmetic
 of course X, Y not independent!
 Experiment 3:
 create random values X and X_{g}
 compute Y = X  X_{g}
 test Y ∼ U(−0.5, 0.5) with χ^{2} test →

N 
pvalue 
3000 
0.6165 
30000 
0.5279 
 compute correlation coefficient ρ(X,Y) →

N 
ρ 
3000 
0.0077 
30000 
0.0085 
300000 
0.0003 
 small, tends to 0 for large N (seemingly)
 Approximate probability density function f_{s}:
 assume X, Y independent
 → computation of density function easy via
convolution
 Gaussian smoothed over interval 1
 coincides with exact discrete distribution at integer
x
 f_{s} and corresponding cdf F_{s}
easily computed numerically
 for standard values used here
 Experiment 4:
 χ^{2} testing using f_{s}
 create rounded values as before
 use edges on integral boundaries → test is
sensitive to rounding
 use F_{s} instead of Φ
 results

N 
pvalue, Φ 
pvalue, F_{s} 
3000 
0.0044 
0.0046 
30000 
3.7098e42 
4.9609e42 
 idea completely useless!