Example
- Example
- 設
為獨立且有相同分布的隨機變數, 其期望值為 ,
變異數為 ,
且令
為樣本均數.
,稱為 離差 (deviation),
因為它們是每一個別資料和樣本均數的差. 令 S2 表示這些離差的平方和.
也就是說,
隨機變數 S2/(n-1) 稱為 樣本變異數 (sample variance).
求 (a)
,
和 (b)
E[S2 /(n-1)].
- Solution:
- (a)
(b)
將上式兩邊取期望值得
其中最後一個等式利用 (a) 的結果. 將兩邊除以 n-1,
得樣本變異數的期望值就是分布變異數
.
- Example (Variance of a Binomial Random Variable)
- Compute the variance of a binomial random variable X with parameters n and
p.
- Solution:
- Since such a random variable represents the number of successes in nindependent trials when each trial has a common probability p of being a
success, we may write
where the Xi are independent
Bernoulli random variables such that
Hence, we have
.
But
and thus