# Questions tagged [order-statistics]

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 inference.

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### Distribution of the minimum among $N$ random variables

I read this post https://stats.stackexchange.com/questions/220/how-is-the-minimum-of-a-set-of-iid-random-variables-distributed where I can find how to compute the density distribution of the minimum ...
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### Probability of the minimun value of n geometric variables be y

I already have the answer to this problem but i would like to understando if and why my resolution is correct or no. Could anyone help me, please? Question: Let $X_1, X_2, X_3, \dots X_n$ be ...
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### Empirical CDF and Order statistics

Screenshot of a result with sketch of its proof I have understood the parts above the red arrow. I'm stuck after that. While I can intuitively understand that the difference between the order ...
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### The expectation of the minimum of two IID binomial distribution Bi(m,1/M) only depends on m/M

I want to prove following proposition: $X,Y \sim Bi(m,1/M)$, $X$ and $Y$ are independent. Prove that $f(m,M)= \mathbb{E} \min(X,Y)$ only depends on $\dfrac{m}{M}$. I have done the following ...
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### Sharp Concentration of General Order Statistics of Gaussian

Suppose $X_{1}, \ldots, X_{n}\overset{i.i.d.}{\sim} \mathcal{N}(0,1)$. Consider the order statistics $X_{(1)} \geq X_{(2)} \geq \ldots \geq X_{(n)}$. Then the Gaussian maxima $X_{(1)}$ has sub-...
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### Distribution of a sum of first and last order statistics

I can't find anywhere on the internet a solution to the following exercise: Let $X = (X_1, \dots, X_n)$ be a sequence of i.i.d random variables from exponential distribution. Find the distribution of ...
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### Unjustified trick for computing difference in order statistics of exponential distribution

It is well known that if $X_1,...,X_n \overset{iid}{\sim} Exp(\lambda)$, then $Y_i:=X_{(i)}-X_{(i-1)} \overset{ind}{\sim} Exp((n+1-i)\lambda)$ for $i=1,...,n$ ($X_{(0)}:=0$). The standard way is to ...
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### Order of appearance of distribution support in an i.i.d sequence

Consider a discrete distribution $\mathcal{D}$ with probabilities $(p_1,\ldots,p_n)$ over numbers $1, \ldots, n$. In a sequence of i.i.d. draws from $\mathcal{D}$, $X_1, X_2, \ldots$, let $X^{(i)}$ ...
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### What is the probability that $\max(X_1, X_2, \ldots, X_n) < \min(Y_1, Y_2, \ldots, Y_n)$ where the $X_i$'s and the $Y_i$'s $\sim U[0, 1]$? [closed]

What is the probability that $\max(X_1, X_2, \ldots, X_n) < \min(Y_1, Y_2, \ldots, Y_n)$ where: $X_i \sim U[0, 1]$ and $Y_i \sim U[0, 1]$, and $X$ and $Y$ are independent continuous random ...
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### Independence among order statistics

Given i.i.d uniform variables $X_1, ..., X_n \sim U[0,1]$, is it true that the event $X_i \leq \min_{k \in [i-1]} X_k$ is independent of $X_j \leq \min_{k \in [j-1]} X_k$ for any $i > j$? The ...
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### Median:Mean::Percentile:?

Here is a simple question that I am hoping has a simple answer. The median and mean are both estimates of the central tendency of a random variable. They both have different advantages in different ...
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### Conditional probability of order statistics

Let $X_1, \ldots, X_n$ be $n$ independent random variables distributed according to some cumulative function $F$. Let $X_{(1)}, \ldots, X_{(n)}$ denote the order statistics. It is a well-known result ...
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### Calculating Probability of a horse winning based on expected position [closed]

My question is this: If you have 4 horses (A , B , C and D ) in a race: Horse A is expected to get position 2 Horse B is expected to get position 2 Horse C is expected to get position 3 Horse D is ...
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### Confidence Interval - Meaning and Interpretations

Could someone please help me understand what exactly "Confidence" in confidence interval actual means? Does it mean, that, (on average or exactly?), we can say, that (for a 90% confidence ...
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### Conditional density of the second highest order statistics

Notation: $Y_1$: Highest order statistics of $(N-1)$ players' valuation. $F_n^M:$ The distribution function of the highest $n$th order statistics of $M$ players. $f_n^M:$ The density of the highest $n$...
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### Expectation of the maximum order statistic of Pareto distribution

Suppose X follows the Pareto Type I given, $$P(X>x) = \Bigl(\frac{\gamma}{x}\Bigl)^\alpha, \quad x\geq \gamma,\; \alpha>0.$$ Then, \begin{align*} P(X=x) &= \frac{d}{dx}P(X\leq x)\\ &= \...
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### Why is $\bar X$ mirrored from zero (additive inversion) after the described operation, and what is its maximum value?

Preface: I am currently preparing for a statistics olympiad qualifiers of the one of my country’s universities. Yesterday I came across one task, in which I do not quite understand the intuition ...
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### Upper bound the even momentum of the squared vector length

I am reading a proof the follow lemma Let $w = \frac{1}{\sqrt{d}} (1, \dots, 1)$. Then, for every unit vector $a (a_1, \dots, a_d)$ and for all $k=0,1,\dots$ we have $E[Q(a)^{2k}] \leq E[Q(w)^{2k}]$ ...
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### Joint density of $k$th upper order statistics [duplicate]

I'm trying to show that for $X_1,...,X_n$ iid with distribution function $F$ which has density $f$, the corresponding order statistics $X_{1,n}>\cdots>X_{n,n}$, the joint density of the $k$th ...