求助一道统计题:一个单位长度的线段被两个任意点(均匀分布)切成三个线段。怎么证明中间的线段是beta分布?(α = 1, β = 2)
一个单位长度的线段被两个任意点(均匀分布)切成三个线段。
- 令X表示中间线段的长度,说明X是一个α = 1, β = 2的贝塔分布
(注:一个α = 1, β = 2的贝塔分布的概率密度函数为:当0 < x < 1时,f(x) = 2(1 − x) ;其他时f(x) = 0)
求 中间线段的长度 小于 剩下两段的长度的算术平均值 的概率
求 中间线段的长度 小于 剩下两段的长度的几何平均值 的概率
A line segment of unit length is cut at two random points to create three pieces.
(i) Let X denote the length of the middle piece. Show that X has a Beta(α = 1, β = 2) distribution.
(ii) What is the probability that the length of the middle piece is less than the (arithmetic) average length of the remaining two end pieces?
(iii) (Bonus) What is the probability that the length of the middle piece is less than the geometric mean of the lengths of the remaining two end pieces? [10+10 (+5) = 20 (+5) points]
Note: The probability density function (pdf) of a Beta(α = 1, β = 2) distribution is given by f(x) = 2(1 − x) if 0 < x < 1 and f(x) = 0, otherwise. Recall, that if a and b are two positive numbers their geometric mean is √ ab and their arithmetic average is (a + b)/2. If you use R for solving (ii) and/or (iii) above, provide your codes as well.
分成两段的感觉还好理解,这样分成三段就没有思路,不知道怎么求他的cdf。