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但 X 1 , X 2 相互独立且都服从 N ( m , s 2 ), 因此 X 2 - X 1 ~ N (0,2 s 2 ), 且

证明题 ：设总体 X ~ N ( m , s 2 ), X 1 , X 2 是取自 X 的样本 ,  X 是样本均值 , S 2 是样本方差 , 试证明  X 与 S 2 独立 , 且 S 2 / s 2 ~ c 2 (1). 证 :. 但 X 1 , X 2 相互独立且都服从 N ( m , s 2 ), 因此 X 2 - X 1 ~ N (0,2 s 2 ), 且.

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但 X 1 , X 2 相互独立且都服从 N ( m , s 2 ), 因此 X 2 - X 1 ~ N (0,2 s 2 ), 且

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  1. 证明题：设总体X~N(m,s2), X1,X2是取自X的样本, X是样本均值, S2是样本方差, 试证明X与S2独立, 且S2/s2~c2(1).证:

  2. 但 X1,X2相互独立且都服从N(m,s2), 因此X2-X1~N(0,2s2), 且

  3. 要证明S2与X相互独立, 只需要证明(X2-X1)与(X2+X1)相互独立即可, 也就是要证明cov(X2-X1,X2+X1)=0, 因为E(X2-X1)=0, 因此 命题成立.

  4. 证明题：设总体X~N(m,s2), X1,X2,X3是取自X的样本, X是样本均值, S2是样本方差, 试证明X与S2独立, 且2S2/s2~c2(2).证:

  5. 定义 下面要证明 (1) Y1和Y2的均值都是0且方差都是s2, (2) Y1,Y2相互独立, (3) Y1和Y2都与X独立. (4) (5)

  6. (1) Y1和Y2的均值都是0且方差都是s2,

  7. (2) Y1,Y2相互独立,因为各样本相互独立且都服从正态分布, 因此Y1,Y2也是正态变量, 正态变量相互独立的充要条件是它们的协方差等于0.

  8. (2) Y1,Y2相互独立,

  9. Y1和Y2都与X独立,

  10. Y1和Y2都与X独立,

  11. (5)

  12. (5) 而Y1~N(0,s2), Y2~N(0,s2)且相互独立, 则 Y1/s~N(0,1), Y2/s~N(0,1), 因此

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