# 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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### Conditional Expectation of RV given the minimum of a Order Statistic

Given an exponential distribution with $X = (X_1,X_2,..,X_n)$ is i.i.d and order statistics $X_{(1)}\le X_{(2)}\le...\le X_{(n)}$, how does one compute $E[X_1|X_{(1)}]$? Intuitively, I know the ...
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### Finding the probability density function of the $n$th largest random variable.

Let $X_1,...,X_{25}$ be independent Unif $[0,1]$ random variables. Let $Y$ be the $13$th largest of the $25$ random variables. Find the probability density function of $Y$. I already know the answer ...
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### Why the median p.d.f. of the uniform distribution is not a p.d.f?

Let $X$ be uniformly distributed on interval $[\theta-2, \theta+2]$, $\theta\in\mathbb{R}$. Let the sample size of $3$, find the p.d.f. of median! I have tried as follows. The p.d.f. of $X$ is \begin{...
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### Prove the CDF of kth Order Statistic

I know the general form of the CDF of kth Order Statistic for n iid random variables is given by $$F_{k}(x) = \sum_{j = k}^{n} {n\choose j}F(x)^j[(1-F(x)]^{n-j}$$ And I am tring to get PDF of the ...
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### Find $E(X_{(1)}\mid T)$ where $T=\sum_{i=1}^n X_i$

Let $X_1,X_2,\ldots,X_n$ be a random sample with $n\geq 2$ from an exponential distribution. $X_{(1)}=\min(X_1,X_2,\ldots,X_n)$. Find $E(X_{(1)}\mid T)$ where $T=\sum_{i=1}^n X_i$. I was able to find ...
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### Correlation coefficients ("Eta, Biserial, Point-Biserial, Gini) [closed]

I want information about the following correlation coefficients. Correlation coefficients ("Eta, Biserial, Point-Biserial, R^2, Gini)
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### Simplifying or approximating $\sum_{k=1}^{\infty}\left(1 - \left(1 - 2^{-k}\right)^n\right)$?

Consider a game in which you flip a coin until you flip tails. Your score is then the number of heads you flipped. So, for example, the sequence $H$, $H$, $H$, $T$ has a score of three, while the ...
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### Mean squared error of order statistic [duplicate]

Let $\theta \in \mathbb{R}$ and $X_1, X_2,..., X_n$ be independent and identically distributed with density \begin{equation} f(x; \theta) = I\left(|x-\theta| \leq \frac{1}{2}\right) \end{...
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### Joint distribution of the $n-1$ nontrivial terms $X_{i}/(\min_{i}X_{i})$ conditional on $\min_{i}X_{i}$

If $X_{i}\sim$Pareto, how can I obtain the joint distribution of the $n-1$ nontrivial terms $X_{i}/(\min_{i}X_{i})$ conditional on $\min_{i}X_{i}$?
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### Correlation Coefficient of two Order Statistics [duplicate]

My problem is exactly the same as asked in here with a change in the notation of the two order statistics. Reframing the question: If $\left(X_1,X_2,…,X_n\right)$ are a random sample from Uniform(0,1)...
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### Picking 5 random numbers on average how big is the largest gap between them?

We pick 5 random numbers from 1 to 100 with repetition. We order them. On average what is the largest difference between two consecutive numbers? What is the smallest difference? Example: 4, 22, 47,55,...
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### Expected value and variance of kth order statistic given maximum value

Let X(1) < X(2) < X(3) < X(4) < X(5) be the order statistics corresponding to a random sample of size 5 from a uniform distribution on [0, θ], where θ ∈ (0, ∞). Prove that the variance of ...