题目

Problem

A discrete random variable $X$ has probability distribution

P(X = x) = \frac{1}{\alpha}, \quad x = 1, 2, \dots, \alpha

A random sample of 6 values is taken from this distribution. The sample values are $8, 7, 3, 7, 2, 9$ .

(a) Show that $2\bar{X}$ is a biased estimator of $\alpha$ .

(3)

(b) Use the sample mean to estimate $\alpha$ .

(2)

题目中文翻译

离散随机变量 $X$ 的概率分布为

P(X = x) = \frac{1}{\alpha}, \quad x = 1, 2, \dots, \alpha

从这个分布中抽取一个大小为 6 的随机样本。样本值为 $8, 7, 3, 7, 2, 9$ 。

(a) 证明 $2\bar{X}$ 是 $\alpha$ 的有偏估计量。

(b) 用样本均值估计 $\alpha$ 。

IAL 2020 Oct S3 Q1