# Calculate one-tailed critical point from a probability distribution

I would like to calculate the numerical value of $\xi\in\mathbb{R}_{\geq0}$$P(|U|<\xi)=0.95$$ where$U\$ is standard-normal distributed.

How may I do that in Mathematica?

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Quantile[NormalDistribution[0, 1], .95] might be what you want? – Daniel Lichtblau Jan 23 '13 at 18:47
@DanielLichtblau Quantile[NormalDistribution[0,1],.975] – ssch Jan 23 '13 at 19:00
@ssch Right you are, I was making a mental reversal somewhere (probably in my mind). – Daniel Lichtblau Jan 23 '13 at 19:02
Dear Kasper, the community has seriously downvoted you on this question because of the way it was asked and the fact that this was something you could have easily looked up in the Mathematica documentation. I have edited your question to be a bit less demanding in tone, and would encourange you to phrase your future questions on Mathematica.SE in a similar way, and to show what you tried. That said, welcome to Mathematica.SE and I hope that you find the site useful in future. – Verbeia Jan 23 '13 at 21:40
@Kasper But then why are you posting homework to a site that is not meant for such? – Daniel Lichtblau Jan 24 '13 at 15:15

Solve[Probability[Abs[u] < \[Xi], u \[Distributed] NormalDistribution[]] == 95/100, \[Xi]]


{{[Xi] -> Sqrt[2] InverseErf[19/20]}

Sqrt[2] InverseErf[19/20]// N


1.959963985

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Including the lower bound gives: Sqrt[2] InverseErf[19/20] – ssch Jan 23 '13 at 18:53
Shouldn't I include Abs[u], like this: Solve[Probability[Abs[u] < [Xi], u [Distributed] NormalDistribution[]] == 95/100, [Xi]] // N – Kasper Jan 23 '13 at 18:58
@Kasper Yes, missed the Abs in the question. – Sjoerd C. de Vries Jan 23 '13 at 19:53
 dist = TransformedDistribution[(Abs[u]), u \[Distributed] NormalDistribution[]];
Quantile[dist .95]
(* 1.95996 *)


or

InverseCDF[dist .95]
(* 1.95996 *)


or

Quantile[NormalDistribution[], .975]
(*  1.95996  *)


or

InverseCDF[NormalDistribution[], .975]
(*  1.95996  *)

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