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Let Z ⌢ Gaussian(0, 1) and X 1, …, X n be a random sample of X = μ + σZ, μ ∈ℝ, σ > 0. Since \(\sqrt {n}\,\frac { \overline X_n-\mu }{\sigma } \frown \text{Gaussian} (0,1)\) and \(\frac {(n-1)\,S_{n}^2}{\sigma ^2}\frown \chi _{(n-1)}^2\) are independent random variables, where \(\overline {X}_n=\tfrac {1}{n} \sum _{k=1}^n X_k\) and \(S_{n}^2 = \tfrac {1}{n-1} \sum _{k=1}^n (X_k- \overline {X}_n)^2\) , the probability density function of Student’s (1908) statistic \(t_{(n-1)}= \sqrt {n}\,\frac {\overline X_n-\mu }{S_{n}}\) is easily obtained.

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Studentized Statistics, Internally and Externally

  • 工具书词条

  • 发表日期：

    2025年

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  • Maria de Fátima Brilhante,
  • Maria Luísa Rocha

摘要

Let Z ⌢ Gaussian(0, 1) and X 1, …, X n be a random sample of X = μ + σZ, μ ∈ℝ, σ > 0. Since \(\sqrt {n}\,\frac { \overline X_n-\mu }{\sigma } \frown \text{Gaussian} (0,1)\) and \(\frac {(n-1)\,S_{n}^2}{\sigma ^2}\frown \chi _{(n-1)}^2\) are independent random variables, where \(\overline {X}_n=\tfrac {1}{n} \sum _{k=1}^n X_k\) and \(S_{n}^2 = \tfrac {1}{n-1} \sum _{k=1}^n (X_k- \overline {X}_n)^2\) , the probability density function of Student’s (1908) statistic \(t_{(n-1)}= \sqrt {n}\,\frac {\overline X_n-\mu }{S_{n}}\) is easily obtained.