If X ¯ and S 2 be the usual sample mean and sample variance based in a random sample of n observation N ( μ , σ 2 ) and T = ( X ¯ − μ ) n S ` prove that C o v ( X ¯ , T ) = σ n − 1 Γ ( n − 2 2 ) 2 n Γ ( n − 1 2 )

Question

If       X    ¯   and       S    2   be the usual sample mean and sample variance based in a random sample of   n observation   N  (  μ  ,      σ          2        ) and   T  =            (              X        ¯            −      μ      )              n              S   ` prove that   C  o  v  (      X    ¯    ,  T  )  =  σ                    n        −        1            Γ              (                              n            −            2                    2                )                            2        n            Γ              (                              n            −            1                    2                )

Lintynx · Accepted Answer

First,                     X        ¯              n   and       S    n   are independent (Basu&#039;s theorem). Then      E                      X        ¯              n    (                    X        ¯              n    −  μ  )      /        S    n    =  Var  ⁡  (                    X        ¯              n    )  ×      E        S    n          −      1        =            σ      2        n    ×            n      −      1                      2            σ                  Γ      (      (      n      −      2      )              /            2      )              Γ      (      (      n      −      1      )              /            2      )      because   σ      S    n          −      1            /        n    −    1    ∼      χ          n      −      1              −      1      . Also note that       E        T    n    =  0. Therefore,  Cov  ⁡  (                    X        ¯              n    ,      T    n    )  =            σ              n        −        1                    2      n                  Γ      (      (      n      −      2      )              /            2      )              Γ      (      (      n      −      1      )              /            2      )        .

If barX and S^2 be the usual sample mean and sample variance based in a random sample of n observation N(mu,sigma^(2)) and T=frac((bar(X)-mu) sqrt(n))(S) prove that Cov(bar(X), T)=sigma (sqrt(n-1)Gamma(frac(n-2)(2)))/(sqrt(2n) Gamma((n-1)/(2))

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