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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13 views

How to find PDF of ordered random variables?

Assumpion: Let $X_1, X_2, \ldots, X_L$ be $L$ independent and identical random variables (RVs). Let $F_{X_i}(x_i)$ and $f_{X_i}(x_i)$ be CDF and PDF of $X_i$. Suppose that $F_{X_i}(x_i) = F_X(x_i)$ ...
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10 views

Extreme value distributions of unaccountably infinite set of random variables

Let us suppose that we have an uncountably infinite set $A=\{x_1,x_2, \cdots\}$ of i.i.d. random variables $x_i$, say with gamma distribution. Are minimum and maximum extreme value distributions ...
1
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32 views

Joint Distribution of k order statistics

For $1\le r_1 < ..<r_k \le n$ integers, I am trying to find the joint pdf of: $$ (X_{(r_1)},...,X_{(r_n)}) $$ where $X_{(r_1)}$ is the $r_1$th largest observation. I am wondering if anyone has ...
0
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1answer
21 views

Dividing by epsilon and taking the limit of epsilon to zero

In deriving the joint distribution of two order statistics, there is the following step (F(x) is the Cumulative Dist Function at x, f(x) is the PDF at x): $$ ...
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22 views

A Question About Probability of ratio of $\max(\cdot)$?

In My field , I reached to this problem. Assumptions: Consider $x_i,\hat{x}_i$ are iid (identical and independent) samples of a joint distribution (e.g., exponential). And also, assume we have $N$ ...
1
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1answer
28 views

Hypothesis testing with order statistics

I know there have been a lot of questions asked on this forum relating to order statistics, so, hopefully, this is not going to be a duplicate. I am trying to understand how I should go about ...
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38 views

Comparing second order statistic of n draws and first order statistic of n-1 draws

Let $x^{(i),n}$ be the $i$-th order statistic of $n$ draws from some distribution function $F$ on $[0,1]$. I have a claim that $$\mathbb{E}[\max \{ x^{(2),n}, m\}] < \mathbb{E}[\max \{ ...
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1answer
36 views

Difference of Ordered Uniform Random Variables

Let $X_1, X_2,..., X_n$ be $n$ random variables distributed uniform(0,1) and $X_{(1)},X_{(2)},..., X_{(n)}$ be the ordered statistics of $X_1,...,X_n$ such that: $X_{(1)} < X_{(2)} < ... < ...
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22 views

Distribution of the maximum of correlated Gaussian variables

I try to unterstand how the distribution of the maximum of a iid Gaussian sample changes when we introduce correlation. For this, I simulated iid random samples and correlated samples with $\sigma =1$ ...
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30 views

Probability of selecting a sequence in order

If "X" number of attempts are made by "Z" number of persons to select a random number from a range "r", where "X <= r". Then I am interested in finding the probability that a particular sequence in ...
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0answers
26 views

Median of the mean/max of an iid sample of exponential variables

I would like to know if one can obtain a simple analytic expression of the median of $T_n$ defined by $$ T_n = \frac{\overline X_n}{X_{(n)}}, $$ where $\overline X_n$ is the empirical mean and ...
2
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1answer
48 views

what does `ensemble average` mean?

I'm studying this paper and somewhere in the conclusion part is written: "Since this rotation of the coherency matrix is carried out based on the ensemble average of polarimetric scattering ...
3
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2answers
38 views

Probability of ordered sequence of Gaussian Distributions

Suppose there are 5 cars racing on a match. Suppose we know that the distributions of their finishing time are roughly in some Gaussian distributions $N(\mu_i,\sigma_i^2), i=1,2,3,4,5$. How can I ...
9
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95 views

Looking for references related to an inequality in order statistics

I was reading the paper "on the minimum of several random variables". In example 10 item (ii) it states: Let $1\leq k\leq n$. Let $g_i,i\leq n$, be independent $N(0,1)$ Gaussian random variables. ...
2
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3answers
43 views

Difference between the largest and second largest observations from a sample of iid normal variables

tl;dr: what can we say about the distribution of s_1 - s_2, where s_1 and s_2 are the largest and second largest draws from a sample of N iid normal variables? "Who's the world's greatest ...
3
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1answer
43 views

Sampling distribution of sample trimmed (truncated) mean

It is elementary probability theory that the sample mean of an i.i.d. sample follows normal distribution, if the background distribution is normal. But what about the trimmed mean? Is there any result ...
2
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1answer
57 views

Probability calculation involving order statistics

There are $n+1$ independent and identically distributed random variables with the same distribution as $D \sim \text{Exp}(\mu)$, denoted by $D, D_1, D_2, \ldots, D_n$. Define event $E_1$ as "$D$ is ...
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2answers
45 views

Probability of being highest OR second highest draw from a distribution

In my situation, I have a distribution F(x) over some compact interval. Say I take $n$ iid draws from the distribution. I want to find the probability that one draw, $x_i$, is the highest of the $n$ ...
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1answer
51 views

Probability ( kth order statistics <x , (k+1)th order stat >x)

consider $n$ iid draws according to some cdf $F$. What is the probability of the following event: the $k^{\text{th}}$ highest value is smaller than $x$ AND the $k+1$ highest value is larger than $x$. ...
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1answer
30 views

Conditional distribution of the sum of k, independent ordered draws

Suppose I make k independent draws from the same distribution (say uniform). The distribution of the sum of these order statistics should equal the distribution of the sum of k independent random ...
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1answer
52 views

Sum of order statistics

Is there a general expression for the pdf of the sum of the order statistics? Suppose they are all drawn independently from the same distribution.
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1answer
30 views

Distribution of non-identical exponential order statistics

This is a problem given as practice for an upcoming exam (without solution provided). Let $X$ and $Y$ be independent exponential random variables, with $X = \theta e^{-\theta x}$ and $Y = \lambda ...
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0answers
18 views

Expected value of k'th largest Gaussian

Suppose $\{g_i\}_{i=1}^n$ are i.i.d. standard normal. Let us sort them in terms of absolute values to obtain $\tilde{g}_i$ where $\tilde{g}_1$ is the largest. Is there a nice (simple,closed form) ...
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0answers
12 views

Empirical likelihood method to Find the order of a parameter given a set of constraints

Firstly we assume that $X_1,...X_n$ are order statistics($X_i\leq X_{i+1}$) from an i.i.d sample of random variables and let $r$ be integer and $r\geq1$. Start with the Equation (1) as following ...
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14 views

(Empirical likelihood method) Find the order of a parameter given a set of constraints

Firstly we assume that $X_1,...X_n$ are order statistics($X_i\leq X_{i+1}$) from an i.i.d sample of random variables and let $r$ be integer and $r\geq1$. Start with the equation (1) \begin{equation} ...
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1answer
47 views

Order statistics when variables have different distributions

Let a,b and c be random variables, where a~U[0,8] and b~U[3,8]. Let c=max{a,b}, What is the mean of c? In general, let $(x_1, ..., x_n)$ be independently distributed in different supports. What is ...
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90 views

Convert min to max probability

Assuming $Y=min(w_1,\ldots,w_n)$ , $w_i\sim N(\mu,\sigma^2) i.i.d$ I want to express $Y$ in terms of the $Q$ function. Knowing that $Y=min(w_1,\ldots,w_n)=-max(-w_1,\ldots,-w_n)$ $P(Y\leq y)= ...
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1answer
47 views

+conditional density of sum of random variables

Let $X$ and $Y$ be two independent random variables distributed uniformly on $[0,1].$ How can I compute the following density: $$f(X+Y|\max\{X,Y\}<a)\ \text{for}\ a\in [0,1]?$$ More generally, ...
2
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2answers
101 views

The maximum and minimum of five independent uniform random variables

Let $U_1,\dots,U_5$ be independent, each with uniform distribution on $(0, 1).$ Let $R$ be the distance between the minimum and maximum of the $U_i^{'}$s. Find the joint density of the max and the ...
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35 views

Probability Solution for Ordered Statistics

I had a question on a solution I saw and the answer is fairly unclear. The question is below and the answer follows it. Let $X_{(1)}\le X_{(2)}...\le X_{(n)}$be the ordered values of n independent ...
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1answer
13 views

fair distribution of table tennis players in roster

Say I have 18 players who have been ranked in order of ability 1-18. I need to make up 6 teams each with 3 players. How do I choose the members of each team such that based on the ranking of the ...
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1answer
28 views

Use of binomial coefficient in order statistics

Appreciate all your help with the following: Let $U_{1}, ... U_{n}$ be n independent and uniformly distributed on (0,1) random variables. Let $U_{(1)} \geq U_{(2)} \geq ... \geq U_{(n)}$ be an ...
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0answers
20 views

showing $E(X_{(i)}| \sum_{i=1}^5 X_i)=1.8E(Z_{(i)})$

suppose 5.5,3.5, 2.5,4.5,2 be a random sample from of gamma distribution with parameters of $ \beta,\alpha=2$. if $Z_{(i)}$ be i-th order statistic a random sample of size 5 from $\Gamma(2,1)$, how ...
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40 views

two-parameter exponential distribution

Now, I have some problems about the distribution of random variable. In my work, let $X_{i}$ for i= 1 to n be iid random variable from two-parameter exponential distribution. We known the Mgf of X is ...
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1answer
53 views

How can I can calculate the CDF of the random variable?

Assuming that $H$, $N$ are random variables in which $H$ is distributed following exponential distribution with mean value $\Omega$, and $N$ is a Gaussian random variable with probability density ...
2
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1answer
50 views

P-value - test at $5 \%$ if there is significant difference in fuel consumption between the two petrol grades?

A car owners want to investigate if gasoline consumption of his car depends on the fuel octane number. He therefore intend to "premium" and "regular" at random and computes each time the ...
1
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1answer
27 views

(bookexercise) sign test of $H_0: \bar m = \bar 25$ against $H_1: \bar m < \bar 25$ there we have 15 known random observation

Follwing data desribes the measured fracture strength of 15 randomly selected units made by a new ceramic material $$20, 42, 18, 21, 22, 35, 19, 18, 16, 20, 21, 32, 22, 20, 24$$ At a previously ...
2
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2answers
70 views

Compute the order statistics probability of $P(X_{(3)} < aX_{(2)})$

Suppose $a>1$, $X_1,X_2, X_3,$ and $X_4$ are four iid continuous random variables with $U(0,a)$ random variables so their common density function is $f(x)=\frac{1}{a}, 0<x<a.$ $a.$ joint ...
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1answer
37 views

determine the confidence level for the confidence interval $I_{\bar m}=(x_{(2)},x_{(9)})$

Let $x_1,...,x_{10}$ be a sample from a continuous random variable $X$ with the median $\bar m$. Determine the confidence level for the interval $I_{\bar m}=(x_{(2)},x_{(9)})$. $x_{i}$ is the ...
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2answers
46 views

True or False — mutual independence and identical distribution? [closed]

Suppose $X_1, X_2,\ldots, X_N$ are mutually independent and identically distributed. Let $F_X$ denote the cdf of $X_1$. Would this statement be true or false? I'm totally lost as to how to approach ...
0
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2answers
62 views

How to find distribution of order statistic

Let $X_i$ be iid random variables with common density $f$ and distribution $F$. Let $Y_k = X_{(k)}$ be the k-th order statistics (that is, $Y_1 = X_{(1)} = \min(X_i)$ etc.). Show that the joint ...
2
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0answers
39 views

Bounding the density of random variable

This is a followup to the question in Bounding the Density of the Maximum of N Random Variables I have a random variable, X, whose cdf is bounded as below: $ \Pr \{X \le x \} \le ...
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2answers
27 views

Equivalence involving expectation

I am stuck with the following problem, where I am asked to prove/disprove the following hypothesis: Is $\mathrm{E}\{e^{\max_i X_i}\} = \mathrm{E}\{\max\limits_i e^{X_i}\}$, where the $X_i$'s are ...
1
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1answer
54 views

Distribution of maximum of correlated Gaussians

Let $X_1,X_2,...,X_n$ be iid standard Gaussian random variables. Consider the set of random variables $M =\left\{\left( X_i-X_j\right) :i,j = \left\{1,2,\dots,n\right\} \& i\ne j\right\}$. I ...
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542 views

Calculate probability of joint distribution pdf and marginal pdf

I came across a research paper where they proceed in the Appendix to calculate the probability: $$P(R_m+R_n>\bar{R}_m+\bar{R}_n)$$ Here $R_m+R_n$ is the non-orthogonal scheme achievable rate and ...
0
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2answers
24 views

Find the Quartile.

There are the following numbers from a sample data: $$36, 45, 49, 53, 55, 56, 59, 61, 62, 65, 69, 71, 76, 78, 81, 85, 91, 92, 99.$$ There are $19$ values. The question is to find the third quartile. ...
2
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1answer
19 views

CDF of middle of $3$ random variables

Let the independent random variables $X_1$, $X_2$, $X_3$ have the same cdf $F(X).$ Let $Y$ be the middle value (second largest) of $X_1$, $X_2$, $X_3$. Determine the cdf of $Y$. (Hint: use a Binomial ...
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0answers
11 views

Family of random vectors with same second-highest order statistics

I am searching for a family of joint distribution of N(N is a finite integer) identical (not necessarily independent) random variables, such that any joint distribution in the family gives the same ...
2
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1answer
189 views

Cramer-Rao and Efficient Estimators

Let $X_1,X_2,X_3,...,X_n$ be a random sample from the exponential distribution having PDF $f(x;\lambda)= \frac{1}{\lambda}e^\frac{-x}{\lambda}\chi\{x>0\}.$ A) Find the Cramer-Rao lower bound ...
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1answer
62 views

Show the consistency of an estimator?

Let $Y_1,Y_2,Y_3,...,Y_n$ be a random sample from the exponential distribution having PDF $f(y;\lambda)= \lambda e^{-y\lambda},$ $y>0.$ A) Show that $\hat\lambda_n = Y_1$ is not consistent for ...