题目
A market stall sells cauliflowers classified by three different weights, small (S), medium (M) and large (L).
The weights, in grams, of these cauliflowers are modelled by independent normal distributions as shown below.
S ~ N(265, 78^2) M ~ N(625, 42^2) L ~ N(886, 45^2)
The random variable T represents the total weight of 2 randomly selected medium cauliflowers minus the weight of 1 randomly selected large cauliflower. T ~ N(a, b)
(a) Find the value of a and the value of b.
(b) Using standardisation, find the probability that a randomly selected large cauliflower weighs more than 3 times the weight of a randomly selected small cauliflower.
A random sample of 2 small cauliflowers is taken.
(c) Using standardisation, find the probability that the first cauliflower in this sample weighs at least 100 grams more than the mean weight of both cauliflowers.
题目中文翻译
某市场摊位出售三种重量等级的花椰菜：小 (S)、中 (M) 和大 (L)。
这些花椰菜的重量（单位：克）分别由相互独立的正态分布建模，如下所示。
S ~ N(265, 78^2)，M ~ N(625, 42^2)，L ~ N(886, 45^2)
随机变量 T 表示 2 个随机选取的中等花椰菜的总重量减去 1 个随机选取的大花椰菜的重量。T ~ N(a, b)。
(a) 求 a 和 b 的值。
(b) 使用标准化方法，求随机选取的一个大花椰菜重量大于随机选取的一个小花椰菜重量 3 倍的概率。
现在抽取 2 个小花椰菜的随机样本。
(c) 使用标准化方法，求样本中第一个花椰菜至少比两者平均重量重 100 克的概率。