A329284 - OEIS

%I #11 Feb 16 2025 08:33:58

%S 3,0,9,0,2,3,2,3,0,6,1,6,7,8,1,3,5,4,1,5,4,0,3,9,9,8,3,0,1,0,7,3,7,9,

%T 2,0,5,4,9,1,0,0,8,4,9,1,8,6,5,8,0,8,8,5,5,6,9,7,1,7,1,1,0,8,5,4,3,5,

%U 6,9,1,4,2,8,9,5,1,4,5,5,5,3,1,2,2,6,6,7,2,4,1

%N Decimal expansion of the quantile z_0.999 of the standard normal distribution.

%C z_p is the number z such that Phi(z) = p, where Phi(x) = Integral_{t=-oo..x} (1/sqrt(2*Pi))*exp(-t^2/2)*dt is the cumulative distribution function of the standard normal distribution. This sequence gives z_0.999.

%C This number can also be denoted as probit(0.999), where probit(p) is the inverse function of Phi(x). See the Wikipedia link below.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuantileFunction.html">Quantile Function</a>

%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Probit">Probit</a>

%e If X ~ N(0,1), then P(X<=3.0902323061...) = 0.999, P(X<=-3.0902323061...) = 0.001.

%o (PARI) default(realprecision, 100); solve(x=0, 5, erfc(x)-2*0.001)*sqrt(2)

%Y Quantiles of the standard normal distribution: A092678 (z_0.75), A329280 (z_0.9), A329281 (z_0.95), A329282 (z_0.99), A329283 (z_0.995), this sequence (z_0.999), A329285 (z_0.9995), A329286 (z_0.9999), A329287 (z_0.99999), A329363 (z_0.999999).

%K nonn,cons,changed

%O 1,1

%A _Jianing Song_, Nov 12 2019