[機統]求 t- distribution 的 variance

[yueayase (scrya)]

t - distribution 可以用兩種方式定義:

(1) Let Z be a standard normal random variable and Y be a chi-square random
    variable with v degrees of freedom. Also, Z and Y are independent.
    Define a random variable
    T = Z / √(Y/v), then the distribution of T is called the t- distribution.


                                                                    2
(2) The distribution density function           Γ((v+1)/2)        t   -(v+1)/2
                                       f(t) =   -------------- (1+---)
                                                √(πv) Γ(v/2)    v


 -∞ < t < ∞ is called the t- distribution.

                       2
如果用(1)去算 E(T), E(T ):

                1/2   -1/2       -1/2
E(T) = E(Z/(Y/v) ) = v    E(Z)E(Y   ) = 0 (因為E(Z) = 0)

   2      2               2     -1
E(T) = E(Z /(Y/v)) = v E(Z ) E(Y )

     2    2           2
因為Z ~ χ  , 所以 E(Z ) = 1
          1

   -1      1
E(Y  ) = ------ (用積分算)
          v-2

所以 Var(T) = v/(v-2)

但是用(2)的話,
        ∞
E(T) = ∫ tf(t) = 0 (因為 tf(t) 是奇函數 )
        -∞

        2                              2
但是 E(T ) 好像不好求(Hint是給: Let 1+t / v = 1/u)

但是這樣一來,積分上下界不都變為0,好奇怪...


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推 Annihilator :f is even...
推 ato0715     :因為(1)式中已經假設Z為normal distribution 了
→ ato0715     :根據中央極限定理，T分配要變成常數分配的話那樣本數
→ ato0715     :n要夠大(最好能到無限大，但實務上約取30個)
→ ato0715     :而在(2)式中，根據積分可得E(T*T)=n/n-2
→ ato0715     :如果n很大，則n/(n-2) 趨近於 1 ，(1)(2)式所得相同
→ ato0715     :這種解釋方法如有錯誤，還請板上先賢們指教