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using Distributions
using Statistics
function monotone_approx(N::Int64,n_trials::Int64,n::Int64)
N: the range of U([-N,N])
n_trials: the number of times we generate random vectors
n: the 'dimension' of the random vector
Q: the quality of the approximation for ten values
monotone relation: a logical verification that the monotone relation holds
X = rand(-1*N:N,n_trials,n);
Y = sum(X,dims=(2));
## comparison with binomial:
Q = zeros(10);
D, D_hat = zeros(10), zeros(10);
for k =1:10
## delta is approximately N/2
delta = N/2
## an approximation of n_hat which is 'n' in the n choose k formula
n_hat = Int(floor((2*N/(2*N+1))*n));
## the fraction of S_n whose value is greater than 2*Delta*k:
D[k] = mean(Y.>2*delta*k);
## k in the n choose k formula:
D_hat[k] = Int(floor((2*k + n_hat)/2));
## a CDF that should have a monotone relationship with W
## this may be considered an approximation
Z = 1-cdf(Binomial(n_hat),D_hat[k]);
## checking the quality of the approximation
Q[k] = min(D[k],D_hat[k])/max(D[k],D_hat[k])
return Q, minimum((D[2:10].-D[1:9]).*(D_hat[2:10].-D_hat[1:9])) > 0
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