题目
Small containers and large containers are independently filled with fruit juice. The amounts of fruit juice in small containers are normally distributed with mean 180 ml and standard deviation 4.5 ml The amounts of fruit juice in large containers are normally distributed with mean 330 ml and standard deviation 6.7 ml The random variable W represents the total amount of fruit juice in a random sample of 2 small containers minus the amount of fruit juice in 1 randomly selected large container. W ~ N(a, b) where a and b are positive constants.
(a) Find the value of a and the value of b
(b) Find the probability that a randomly chosen large container of fruit juice contains more than 1.8 times the amount of fruit juice in a randomly chosen small container.
A random sample of 3 small containers of fruit juice is taken.
(c) Find the probability that the first container of fruit juice in this sample contains at least 5 ml more than the mean amount of fruit juice in all 3 small containers.
题目中文翻译
小容器和大容器分别独立装有果汁。 小容器中的果汁量服从正态分布，均值为 180 ml，标准差为 4.5 ml。 大容器中的果汁量服从正态分布，均值为 330 ml，标准差为 6.7 ml。 随机变量 W 表示随机抽取的 2 个小容器中果汁总量减去 1 个随机选取的大容器中的果汁量。 W ~ N(a, b)，其中 a 和 b 为正实数常数。
(a) 求 a 的值和 b 的值。
(b) 求随机选取的一个大容器中的果汁量超过随机选取的一个小容器中果汁量的 1.8 倍的概率。
抽取 3 个小容器作为随机样本。
(c) 求该样本中的第一个容器的果汁量至少比 3 个小容器平均果汁量多 5 ml 的概率。