题目
A bag contains 10 counters each with exactly one number written on it.
There are 6 counters with the number 7 on them There are 3 counters with the number 8 on them There is 1 counter with the number 9 on it
A random sample of 3 counters is taken from the bag (without replacement). These counters are then put back in the bag. This process is then repeated until 20 samples have been taken. The random variable represents the number of these 20 samples that contain the counter with the number 9 on it.
(a) (i) Find the mean of
(ii) Find the variance of
A random sample of 3 counters is chosen from the bag (without replacement). (b) List all possible samples where the median of the numbers on the 3 counters is 7 (c) Find the sampling distribution of the median of the numbers on the 3 counters.
(Total for Question 7 is 12 marks)
题目中文翻译
一个袋子里有 10 个计数器，每个计数器上都写着一个数字。
其中有 6 个写着 7，3 个写着 8，1 个写着 9。
从袋子中抽取 3 个计数器（不放回）。 然后把这些计数器放回袋中。 这一过程重复进行，直到共抽取了 20 个样本。 随机变量 表示这 20 个样本中含有写着 9 的计数器的样本个数。
(a) (i) 求 的均值。
(ii) 求 的方差。
从袋中再随机抽取 3 个计数器（不放回）。 (b) 列出所有 3 个计数器上的数字中位数为 7 的可能样本。 (c) 求这 3 个计数器的中位数的抽样分布。
（第 7 题共 12 分）