0
votes
1answer
141 views

Sufficient statistic for uniform distribution

Given random sample $\left\{ { X }_{ 1 },{ X }_{ 2 },...,{ X }_{ n } \right\} $ from $ U(0,\theta)$. Let ${Y}_{i}$ be the order statistics. Then the sufficient statistic for $\theta$ is ${ Y }_{ n ...
0
votes
0answers
44 views

If $n$ order stats are iid from Uniform(0,1), why does dividing by the highest order stat give $n−1$ order stats iid from Uniform(0,1)?

As the title states: If $P_{(1)}, ... ,P_{(n)}$ are order statistics of $n$ independent uniform $(0,1)$ random variables, why are $P_{(1)}/P_{(n)} ..... P_{(n-1)}/P_{(n)}$ also order statistics of ...
0
votes
1answer
372 views

Range of Uniform Distribution

I'm trying to compute the density for the range $R_n$ for samples of a random variable $X$ that are uniformly distributed on the interval $(a,b)$. We define the range as $$ R_n = X_{(n)} - X_{(1)}, ...
1
vote
1answer
270 views

Order statistic and independence

"If $Y_1, Y_2, ..., Y_n$ are independent, uniformly distributed random variables on the interval $[0, \theta]$, show that $U=Y_{(1)}/Y_{(n)}$ and $Y_{(n)}$ are independent." I have already found ...
1
vote
0answers
71 views

Approximation or calculation of the probability of getting “clumps” when sampling from a uniform distribution

Suppose that there are $n$ independent samples $X_1,X_2,...,X_n$ sampled from the uniform distribution on $[0,1]$ with the pdf $f(x)=1$. Is there a good way to calculate or approximate the ...