Confidence interval of x random points in ND given size comparison Say we have a standard normal distribution, in which a random sample x(0) is selected. We know the 95% confidence interval is simply [-2, 2] by definition.


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*Let's say we now select another random sample, x(1), and we're told x(1) > x(0). What is the 95% confidence interval of x(0) now?

*Let's say we have x(0..n) and we know x(i) > x(i-1) (for 1 <= i <= n | i in Z obviously), what would be the general equation to compute the confidence interval of x(i)?
(Optional - answer for 2 is adequate - this is just for curiosity..)
3. What if we have x(i) >= x(i-1) rather than x(i) > x(i-1)?
 A: I would not use "confidence interval" in this sense, but I assume you mean to ask what interval contains $95\%$ of the posterior probability distribution of $x_{0}$.
There is a $2.5\%$ probability that the lowest of sample size $n$ from a uniform distribution on $[0,1]$ is below $1-(1-0.025)^{1/n}$ and a $97.5\%$ probability it is below about $1-(1-0.975)^{1/n}$.  So if $\Phi(x)$ is the cumulative distribution function of a standard normal, then your interval could be between $\Phi^{-1}\left(1-(1-0.025)^{1/n}\right)$ and $\Phi^{-1}\left(1-(1-975)^{1/n}\right)$ and you can slightly simplify this to the interval between $-\Phi^{-1}\left(\sqrt[n]{0.975}\right)$ and  $-\Phi^{-1}\left(\sqrt[n]{0.025}\right)$.  
There are other intervals with $95\%$ probability.  This particular one has the merit of having the same probability of being above the interval of being above it is the same as being below it.
As for actual values, the following R code does some calculations and is easily adaptable to find intervals for other probabilities or other positions in the sample.
interval <- function(sample_size, position=1, prob=0.95){
  c(qnorm(qbeta((1-prob)/2, shape1=position, shape2=sample_size-position+1)),
    qnorm(qbeta((1+prob)/2, shape1=position, shape2=sample_size-position+1)))}

and gives the following values
> interval(sample_size=1)
[1] -1.959964  1.959964
> interval(sample_size=2)
[1] -2.238964  1.002240
> interval(sample_size=3)
[1] -2.3908915  0.5463818
> interval(sample_size=4)
[1] -2.4943466  0.2594725
> interval(sample_size=5)
[1] -2.57233441  0.05473134
> interval(sample_size=6)
[1] -2.6346870 -0.1023029

with the first interval being the familiar $95\%$ interval for the normal distribution and the second being the answer to your question $1$.
