OFFSET
1,1
COMMENTS
The above data has only 51 entries because the Mathematica code below would give numerically unstable results for the integrals with n>51.
LINKS
Matt Frank, Answer to "Numerical superiority necessary to beat in L^oo a population one standard deviation ahead", on CrossValidated (aka stats.stackexchange.com)
EXAMPLE
For n = 2, a(2) = 11 because if X_i are standard normal variables, and Y_j are normal variables with mean and variance 1, then P[max(X_1, ... X_10) > max(Y_1, Y_2)] = 49% but P[max(X_1, ... X_11) > max(Y_1, Y_2)] = 51%.
MATHEMATICA
cdf[x_] := CDF[NormalDistribution[], x];
prob[m_] := NIntegrate[D[cdf[x]^m, x] cdf[x - 1]^n , {x, -Infinity, Infinity}]
Table[FindArgMin[(prob[m] - 1/2)^2, {m, n^1.75}] // Ceiling, {n, 51}]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Matt Frank, Nov 03 2021
STATUS
approved