题目
Problem
The time, in minutes, spent waiting for a call to a call centre to be answered is modelled by the random variable with probability density function
\dfrac{1}{192}(t^3 - 48t + 128) & 0 \leqslant t \leqslant 4 \\ 0 & \text{otherwise} \end{cases}$$ (a) Use algebraic integration to find, in minutes and seconds, the mean waiting time. (b) Show that $P(1 < T < 3) = \dfrac{7}{16}$ <div style="text-align: right;">(3)</div> <div style="text-align: right;">(3)</div> A supervisor randomly selects 256 calls to the call centre. (c) Use a suitable approximation to find the probability that more than 125 of these calls take between 1 and 3 minutes to be answered. <div style="text-align: right;">(5)</div> (Total for Question 2 is 11 marks)题目中文翻译
等待呼叫中心接听电话所花费的时间（单位：分钟）用随机变量 建模，其概率密度函数为
\dfrac{1}{192}(t^3 - 48t + 128) & 0 \leqslant t \leqslant 4 \\ 0 & \text{otherwise} \end{cases}$$ (a) 使用代数积分求平均等待时间，答案用分钟和秒表示。 (b) 证明 $P(1 < T < 3) = \dfrac{7}{16}$。 随机抽取 256 个呼叫。 (c) 使用适当的近似方法，求这些呼叫中超过 125 个的等待时间介于 1 到 3 分钟之间的概率。 （第 2 题共 11 分） </details> # 解答