容量分別是n1,n2.組成一個聯合樣本n1+n2,組合樣本的方差是?

題目：

容量分別是n1,n2.組成一個聯合樣本n1+n2,組合樣本的方差是?
從總體中抽取兩組樣本，其容量分別爲n1與n2，設兩組的樣本均值分別爲 X1與X2,樣本方差分別爲S1^2及S2^2,把這兩組樣本合併爲一組容量爲n1+n2的聯合樣本，

解答：

組合樣本的方差是 S1^2 + S2^2
This problem is not as simple as the answer suggests here. When combining m random variables: n1, n2, ..., nm each with an average value of X1, X2, ..., Xm and a standard deviation S1, S2, ..., Sm, the average of the new random variable N = n1 + n2 + ... + nm will have an average value of X = X1 + X2 + ... + Xm, this is expected. The standard deviation of the new random variable, though, is S = (S1^2 + S2^2 + ... + Sm^2)^(1/2). So the variance of the new random variable or Var(N) = S1^2 + S2^2 + ... + Sm^2
再問： 不是很明白，不是該用樣本方差的定義來做嗎？ S = (S1^2 + S2^2 + ... + Sm^2)^(1/2) 這個結論是怎麼得到的？怎麼證明？