[機統]求 t- distribution 的 variance
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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