题目
Problem
The distance, in metres, a novice tightrope artist, walking on a wire, walks before falling is modelled by the random variable W with cumulative distribution function
0 & w<0 \\ \dfrac{1}{3}\left(w-\dfrac{w^4}{256}\right) & 0 \leqslant w \leqslant 4 \\ 1 & w>4 \end{cases}$$ (a) Find the probability that a novice tightrope artist, walking on the wire, walks at least 3.5 metres before falling. <div style="text-align: right;">(2)</div> A random sample of 30 novice tightrope artists is taken. (b) Find the probability that more than 1 of these novice tightrope artists, walking on the wire, walks at least 3.5 metres before falling. <div style="text-align: right;">(3)</div> Given E(W) = 1.6 (c) use algebraic integration to find Var(W) <div style="text-align: right;">(5)</div> (Total for Question 2 is 10 marks)题目中文翻译
一名在钢丝上行走的新手走绳艺人，在跌落之前所走的距离（单位：米）由随机变量 建模，其累计分布函数为
0 & w<0 \\ \dfrac{1}{3}\left(w-\dfrac{w^4}{256}\right) & 0 \leqslant w \leqslant 4 \\ 1 & w>4 \end{cases}$$ (a) 求一位新手走绳艺人在跌落前至少走 3.5 米的概率。 (b) 随机抽取 30 位新手走绳艺人。求其中多于 1 人在跌落前至少走 3.5 米的概率。 已知 $E(W)=1.6$。 (c) 使用代数积分求 $\mathrm{Var}(W)$。 （第 2 题共 10 分） </details> # 解答