In probability and statistics, studentized range distribution is the continuous probability distribution of the studentized range of an i.i.d. sample from a normally distributed population. Suppose that we take a sample of size n from each of k populations with the same normal distribution N(μ, σ2) and suppose that y ¯ min {\displaystyle {\bar {y}}_{\min }} is the smallest of these sample means and y ¯ max {\displaystyle {\bar {y}}_{\max }} is the largest of these sample means, and suppose s² is the pooled sample variance from these samples. Then the following statistic has a Studentized range distribution.
q = y ¯ max − y ¯ min s / n {\displaystyle q={\frac {{\overline {y}}_{\max }-{\overline {y}}_{\min }}{s/{\sqrt {n\,}}}}}
Definition
Probability density function Differentiating the cumulative distribution function with respect to q gives the probability density function.
f R ( q ; k , ν ) = 2 π k ( k − 1 ) ν ν / 2 Γ ( ν / 2 ) 2 ( ν / 2 − 1 ) ∫ 0 ∞ s ν φ ( ν s ) [ ∫ − ∞ ∞ φ ( z + q s ) φ ( z ) [ Φ ( z + q s ) − Φ ( z ) ] k − 2 d z ] d s {\displaystyle f_{\text{R}}(q;k,\nu )={\frac {{\sqrt {2\pi \,}}\,k\,(k-1)\,\nu ^{\nu /2}}{\Gamma (\nu /2)\,2^{\left(\nu /2-1\right)}}}\int _{0}^{\infty }s^{\nu }\,\varphi ({\sqrt {\nu \,}}\,s)\,\left[\int _{-\infty }^{\infty }\varphi (z+q\,s)\,\varphi (z)\,\left[\Phi (z+q\,s)-\Phi (z)\right]^{k-2}\,\mathrm {d} z\right]\,\mathrm {d} s}
Note that in the outer part of the integral, the equation
φ ( ν s ) 2 π = e − ( ν s 2 / 2 ) {\displaystyle \varphi ({\sqrt {\nu \,}}\,s)\,{\sqrt {2\pi \,}}=e^{-\left(\nu \,s^{2}/2\right)}}
was used to replace an exponential factor.
Cumulative distribution function The cumulative distribution function is given by
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