The order statistics of a sample are the values placed in ascending order. The i-th order statistic of a statistical sample is equal to its i-th smallest value; so the sample minimum is the first order statistic & the sample maximum is the last. Order statistics are widely used in non-parametric ...

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4
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78 views

Conditional expected value of mutlitple draws from uniform distribution

There are $m$ i.i.d. draws of $x$ made from a uniform distribution on $[0,1]$. The $n$ ($n\leq m$) lowest draws are "winners", i.e. if we write $x_1\leq\ldots\leq x_n\ldots\leq x_m$, the draws $x_1$ ...
0
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0answers
12 views

Calculating the mean of the order statistics of Rayleigh random variables

I am trying to compute the mean of the ith order statistics for $n$ Rayleigh random variables as follows: $\int_0^\infty A \, x \, F^{i-1} (1-F)^{n-i} \, f \, dx$ where $A = i \binom{n}{i}$ and $F=1-...
1
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1answer
22 views

For a combined order statistics involving 2 sample, How would you find the probability (0<MedX<MedY)?

Suppose that there are two samples $X_1$,$X_2$,...,$X_n$ and $Y_1$,$Y_2$,...,$Y_n$. An order Statistics is obtained by combining the two samples and arranging them in increasing order. Then how would ...
6
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1answer
85 views

Order statistics for discrete uniform random variables

Let $X_i, i=1,\cdots,N$ be i.i.d. discrete uniform random variables, taking values in the range $\{0,1,...,M-1\}$. Let $X_{(i)}$ denote the $i$-th order statistic. What are the values of $\...
0
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2answers
42 views

Order Statistics Intuition

I'm having trouble understanding the intuition behind the answer to a practice question. An internet company has three redundant servers for its web site. Thus, the site functions properly as long as ...
0
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0answers
8 views

Operations on two normal distributions using order statistics

$G(x)$ is a Normal distribution with mean $\mu$ and standard deviation $\sigma$. I observe realization of $X$ which are a function of $s$. The distribution $F(s)$ is found as the root (between 0 and ...
0
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0answers
17 views

Compute completion time using order statistics

I am trying to figure out how order statics work, I need to estime the completion time of a computation. The computation consists of n stages and runs at N threads (parallel computing). Assuming that ...
0
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1answer
25 views

Relation between estimator's consistency and biasedness

I have two quick question: If an estimator is consistent, does that imply it is unbiased? If an estimator is biased, does that imply it is not consistent? we know that consistency means ...
0
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0answers
21 views

Covariance of order statistics

I'm a researcher in social science and I have encountered the following math formulation of a problem in my field. Let $x_1,x_2,...,x_n,x_{n+1}$ be $n+1$ i.i.d. random variable with non-negative ...
1
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0answers
36 views

Statistics $X_{(1)}$ complete for a Uniform Distribution?

Someone had asked this earlier, but since it was good practice for my qualifying exam coming up, I figured I would ask and share my work on the problem. The problem is: Suppose $X$ is Unif$(0,\...
2
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1answer
26 views

Use cdf to find expectation

I have a cdf for a $\mathbf {discrete}$ random variable, $X$, $$F_X(x)=1-(1-p)^{xn}$$ where $p\in(0,1)$, $n\in\mathbb N$, $x\in\mathbb N$ My thought is to use $$E[X]=\sum_{x=0}^\infty (1-F_X(x)...
0
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0answers
29 views

Probability with ordered statistics and exponential distribution involved

Assume that $X_1,X_2$ are independent random variables with exponential distribution with the same mean 100. Let $X_{(1)}=\min\{X_1,X_2\}$ and $X_{(2)}=\max\{X_1,X_2\}$. Calculate $P(e^{-0.01X_{(1)}}+...
1
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1answer
40 views

Uniform Distribution Estimator (not MLE)

Does anyone know where the estimator $\hatθ = X _{(1)} + X_ {(n)}$ for a U(0, θ) distribution comes from? Where: $X _{(1)}$ = min$_i (X_i)$ $X_ {(n)}$ = max$_i (X_i)$ I know it is not the MLE, ...
1
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1answer
59 views

Order Statistics with two Groups of Draws

Let $X_{1},X_{2},\ldots,X_{m},\ldots,X_{n}$ be independently drawn from a distribution $F$ and let $Y_{k}^{(n)}$ be the $k$-th order statistic (Convention: $Y_{1}^{(n)}>Y_{2}^{(n)}>\cdots>Y_{...
1
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0answers
22 views

Reshuffling the order statistic of uniform at midpoint

Let $U_1,\dots,U_n$ be i.i.d. drawn from the uniform distribution on $[0,1]$ and let $U_{(1)} \le U_{(2)} \le \dots \le U_{(n)}$ be their order statistics. Assume that $n$ is even and that we take $U_{...
0
votes
0answers
10 views

Distribution of sample minimum after bivariate selection (double truncation)

Let $X$ and $Y$ be two RVs with joint distribution $$ (X,Y)\sim \text{Normal}(\mu,\Sigma) $$ Suppose that there is selection on $X$ and $Y$, such that we observe a vector of realisations of $X$, $\{...
0
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0answers
14 views

Analytical expression for expected the n-th and (n-1)th order statistic of minimum of two uniform random variables

I have two independent random variables $X_1 \sim Uniform(b,1)$ and $X_2 \sim Uniform(0,1)$, where $0<=b<1$. Their CDF's are: $$ F_{X_1}(z) = \begin{cases} 0 & z<b \\ \frac{z-b}{1-b} &...
0
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1answer
25 views

Probability and expectation of three ordered random variables

I am really stuck on the following question during my exam preparation. Let $X_1$, $X_2$ and $X_3$ ...
0
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0answers
11 views

Distribution of a single order statistic (descending order)

I am looking for the distribution of single order statistic ($F_{(r)}(x)$, i.e., the CDF of $X_r$) that is in the form $X_1\geq X_2 \geq \cdots X_r \geq \cdots \geq X_n$. All $X$ are independent and ...
0
votes
1answer
21 views

Joint distribution of largest and last sample

Suppose 12 i.i.d. integer samples are taken from 1 to 10 with uniform probability. What is the joint distribution of the 3 last numbers ($x_{10},x_{11},x_{12}$) and the 3 largest numbers ($\max_1(x_1,....
1
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1answer
44 views

Density of maximum of order statistics

Let $X_{1},X_{2},\ldots,X_{n}$ be independently drawn from a distribution $F$ and let $Y_{k}^{(n)}$ be the $k$-th order statistic (Convention: $Y_{1}^{(n)}>Y_{2}^{(n)}>\cdots>Y_{n}^{(n)}$). ...
1
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1answer
20 views

Marginal Order Statistics

I'm trying to find the marginal statistic of 2 order stats from the full joint and I can't figure out the bounds on the following problem. Consider $0\leq Y_1\leq Y_2\leq Y_3\leq Y_4 < 1$ from the ...
1
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2answers
45 views

An Estimator Based on Exponential RVs

Let $X_1$, $X_2$, $\cdots$, $X_n$ be $n$ random variables independently sampled from the exponential distribution $\text{exp}(1)$. Suppose $k \leq n$, and $X_{(k)}$ is the $k$-th order statistic, i.e.,...
1
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0answers
48 views

upper bound of the expectation of the maximum of a bunch of random variables

I met one technical problem in my project, and I wish some reference here.. Suppose $\{X_i\}_{i=1}^n$ are independent random variables. Are there some inequalities that give upper bound of $\mathbb{E}...
2
votes
1answer
57 views

On expectation of maximum of gaussians

Let $X_1,\ldots,X_n$ be i.i.d $\mathcal{N}(0,1)$ random variables. I am trying to prove that \begin{align} (a)\ \ \mathbb{E} \left[ \max_{i}X_i\right] & \asymp\mathbb{E} \left[ \max_{i}|X_i|\...
2
votes
4answers
28 views

Density function of ordered statistic of a uniform distribution from $0<t<\theta$

Assume $X_1, X_2, ..., X_n$ is a random sample of a uniform random variable $X$ on the interval $(0,\theta)$. What is the density function for $X$ and the density function for $X_{(n)}$ where $X_{(n)}$...
2
votes
0answers
35 views

Ratio of Order Statistics goes to infinity?

Is there a sequence of distributions $(F_n)$ with $F_n(0)=0$ such that $\frac{E[\max(X_{1,n},X_{2,n})]}{E[X_{1,n}]} \to +\infty $ as $n\to +\infty$, where $X_{1,n}$ and $X_{2,n}$ are independently ...
1
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1answer
19 views

Correlation between a random variable and its rank

Let $X_1,\ldots,X_n$ be a random sample from $U(0,1)$ and $X_{(1)}<\ldots<X_{(n)}$ be the corresponding order statistics. Define, $$ R(X_1) = r\quad \text{if}\quad X_{(r)} = X_1;\quad r = 1(1)...
1
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0answers
29 views

Find the distribution of $ \sum_{i=1}^{4} Y_i $, the first four order statistics of five exponential RVs.

The Problem: Consider $X_1, ..., X_5$, $5$ iid, exponentially distributed random variables with $\lambda = 1$. Let $Y_1 \lt Y_2 \lt Y_3 \lt Y_4 \lt Y_5$ be their order statistics. Find the ...
5
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2answers
49 views

What proportion of my friend's calzones were the best he had made so far?

(True story:) A friend of mine was making a calzone and said it wasn't the best he had ever made. It got me wondering, what fraction of the calzones he has made were the best he had made at the time? ...
1
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1answer
32 views

Showing $X_{(n)}$ is an unbiased and consistent estimator for $\theta$.

Let $X_1,X_2,\ldots, X_n$ is distributed iid $\mathrm{Uniform}(0, \theta)$ with $\theta$ in being real positive. Show $\frac{n+1}{n}X_{(n)}$ is an unbiased and consistent estimator for $\theta$. I ...
0
votes
2answers
55 views

Modeling “Time Until Event Occurs”

I'm having trouble understanding why "the time until an event occurs" is modeled as: $S(t) = P[T > t]$ where $T$ is time until event and $t$ is some time. I don't see how $P[T<t]$ or $P[T=t]$ ...
6
votes
3answers
167 views

Are there order statistics for a Gaussian variable raised to a power?

Let $X$ be a random variable with a standard normal distribution. Let $Y = |X|^{2p}$. I am trying to find the distribution for $Y_{(n)}$, i.e., the largest value of $Y$ out of $n$ samples. I have ...
0
votes
2answers
43 views

Probability distribution of the difference of two uniform variables

What is the PDF or CDF of $|X_1 - X_2|$ (absolute value of the difference of two variables). When both $X_1$ and $X_2$ has different Uniform distribution. Note: We can look at the as the distribution ...
2
votes
0answers
23 views

continuity of the sort operator

here's my question. Let $\mathcal{S}\colon\mathbb{R}^n\rightarrow \mathbb{R}^n$ be the function that map a real vector into a vector having its components ordered either way, ascending or descending. ...
0
votes
1answer
54 views

How to find this integral

I want to fin the following integral: $$\int_o^{x_n}\int_0^{x_{n-1}}\cdots\int_0^{x_2}n!\lambda^ne^{-\lambda(x_1+x_2+\cdots+x_n)}dx_1dx_A\cdots dx_{n-1}$$ I think that the answer is $\lambda ne^{-\...
-2
votes
2answers
88 views

Is Schoolyard Team Picking Fair?

Two captains are picking teams for dodgeball. They both will pick players in order of their skill level, best to worst. John gets first pick, Sarah gets the next two picks, and John gets the 4th. If ...
0
votes
0answers
22 views

Median of the sample median equal to the median of the distribution?

I'm taking a course in mathematical statistics this semester. I don't know how much consensus there is among mathematicians about notations and definitions, so I'll just define the essentials. Let me ...
0
votes
0answers
23 views

Probabilities of each waitlist person

Coming from this, $10$ Applicants for a exclusive club membership. I found that you can use total probability to consider existing and old members leaving/returning as members in the club, ended up ...
0
votes
1answer
32 views

Find the probability that the range of a random sample of size $3$ from the uniform distribution is less than $0.8$.

What do i have to use to calculate the probablity that the range of a random sample of size $3$ from the uniform distribution is less than $0.8$? I have the pdf of the range : $$f_W(w)= n(n-1) \...
0
votes
3answers
53 views

Find the joint distribution of the continuous order statistics?

Take $n$ independent variables, ${X_1, X_2,\dots, X_n}$, which are uniformly distributed over the interval $(0,1)$. Then, introduce the variable $M=\min(X_1, X_2,\dots, X_n)$ and the variable $N=\max(...
3
votes
1answer
115 views

The expected value of the $k$th order statistic of iid geometrically distributed rvs, and its asympotic expansion.

I have read the paper Combinatorics of geometrically distributed random variables: Left-to-right maxima. In the paper, the largest order statistic $X_{n:n}$ (i.e., $\max\{X_1,X_2,\ldots,X_n\}$) is ...
1
vote
1answer
64 views

expected value of order statistic

I am having troubles with solving this task 7.4.5. Show that the first order statistic $Y_1$ of a random sample of size $n$ from the distribution having pdf $$f(x;\theta) = e^{-(x-\theta)}, \...
1
vote
0answers
24 views

What is the asymptotic behaviour of $\max_{n \in \{1, 2,\, \dots N\}} \tan n$?

What is the asymptotic behaviour of $f(N) = \max_{n \in \{1, 2,\, \dots N\}} \tan n$? Any non-trivial bounds above or below would be of interest. Some quick numerical experimentation shows: ...
1
vote
3answers
73 views

For $n \geq 2$, let $X_1,X_2,\ldots,X_n$ be independent samples from $P_{\theta}$, the uniform distribution $U(\theta,\theta +1),\theta \in \mathbb R$

For $n \geq 2$, let $X_1,X_2,\ldots,X_n$ be independent samples from $P_{\theta}$, the uniform distribution $U(\theta,\theta +1),\theta \in \mathbb R$. Let $X_{(1)},X_{(2)},\ldots,X_{(n)}$ be order ...
0
votes
0answers
16 views

Given $X_i \sim Geom(p)$, what is the PDF of $R := \min \{ r:\sum_{j=1}^n \mathbb{1}_{\{ X_j \leq r\}} \geq m \}$?

I am considering a file transfer problem that given a file divided into $n$ blocks, the number of transmission rounds of each block $X_i$ satisfies $X_i \sim Geom(p)$. Thus the number of transmission ...
2
votes
1answer
55 views

Show that $(\frac{S_1}{S_n+1},\frac{S_2}{S_n+1},…\frac{S_n}{S_n+1})=_d (U_{(1)},U_{(2)},…,U_{(n)})$.

Let $(X_1, X_2,...,X_n) \in \mathbb R^n$ have density function $p(x)$. (1) Find the density of $(U_{(1)},U_{(2)},...,U_{(n)})$, the order statistics from a sample of iid $\mathbb U[0,1]$ (uniform ...
0
votes
0answers
69 views

subsample order statistics

I am interested in characteristics depending of the $r$-th order statistics $X_r$ of a distribution with unknown pdf. For example, I would like to estimate the Gini mean difference $E(|X_r - X_r'|)$, ...
0
votes
1answer
62 views

compare sample mean and sample median as estimators of µ

Under symmetry, $F^{−1} (0.5) = E(X)$. Compare the sample mean as an estimator of µ to that of the sample median ($F^{−1} (0.5)$) for n sufficiently large, assuming that $X_i$ ∼ Z = N(0, 1), i = 1,...
0
votes
0answers
47 views

Distribution of Interquartile Range on $X_i$ ∼ U(0, 1), i = 1, . . . , 20, iid.

Let Xi ∼ U(0, 1), i = 1, . . . , 20, iid. IQR = $F^{−1}(.75)−F^{−1}(.25)$ = $X_{(15)}−X_{(5)}$ in this example as n = 20. a. Find the distribution of the random variable W = IQR. b. Devise ...