Editing Prediction interval (section)

==== Unknown mean, known variance ====
Given<ref>{{Harvtxt|Geisser|1993|p=[https://books.google.com/books?id=wfdlBZ_iwZoC 7–]}}</ref> a normal distribution with unknown mean ''μ'' but known variance <math>\sigma^2</math>, the sample mean <math>\overline{X}</math> of the observations <math>X_1,\dots,X_n</math> has distribution <math>N(\mu,\sigma^2/n),</math> while the future observation <math>X_{n+1}</math> has distribution <math>N(\mu,\sigma^2).</math> Taking the difference of these cancels the ''μ'' and yields a normal distribution of variance <math>\sigma^2+(\sigma^2/n),</math> thus
:<math>\frac{X_{n+1}-\overline{X}}{\sqrt{\sigma^2+(\sigma^2/n)}} \sim N(0,1).</math>
Solving for <math>X_{n+1}</math> gives the prediction distribution <math>N(\overline{X},\sigma^2+(\sigma^2/n)),</math> from which one can compute intervals as before. This is a predictive confidence interval in the sense that if one uses a quantile range of 100''p''%, then on repeated applications of this computation, the future observation <math>X_{n+1}</math> will fall in the predicted interval 100''p''% of the time.

Notice that this prediction distribution is more conservative than using the estimated mean <math>\overline{X}</math> and known variance <math>\sigma^2</math>, as this uses compound variance <math>\sigma^2+(\sigma^2/n)</math>, hence yields slightly wider intervals. This is necessary for the desired confidence interval property to hold.