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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10 views

orderstatistics of uniform distributions on different ranges

During a simulation I discovered an interesting phenomenon: Given you have 3 agents. 2 are uniformly distributed between [0,1] and one between [0,2]. The question is how often do the smaller agents ...
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2answers
22 views

Order Statistics Expected Value

Can someone help me with this question? I have arrived with the distribution function equal to $g(y) = \frac{n}{\theta }\left(1-\frac{y}{\theta }\right)^{n-1}$. But couldn't solve the integral for the ...
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1answer
24 views

Common distribution of order statistics

Let $X_1,\dots,X_n$ be $iid$ with distribution function F and $Y_1,\dots,Y_n$ $iid$ with distribution function G. I've proved for some function $g$ that $X_1$ is equal in distribution to $g(Y_1)$, ...
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10 views

Order Statistics interval sizes

Suppose an i.i.d. sample of size $n \geq 2$ drawn from a known distribution with density $g$. Let us note the associated order statistics as $(X_{(1)}, \ldots ,X_{(n)})$. I am interested in the number ...
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1answer
19 views

Maximum Likelihood Estimator where x>= theta

Let X = X1, X2, . . . , Xn be i.i.d. random variables from a population with a density if x ≥ θ f(x;θ) = 2θ^2/x^3, if x ≥ θ, 0 elsewhere. What is the Maximum Likelihood Estimator of θ? Is it unbiased? ...
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9 views

Cumulative minimum of an Ornstein-Uhlenbeck process

Assume we generate a sample path $X_t$ from an Ornstein-Uhlenbeck distribution (i.e. a mean-reverting random walk), where $dX_t = −\rho(X_t − \mu)dt + \sigma dW_t$. For concreteness, take $\mu = 0$, ...
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1answer
42 views

The probability of observing a value as extreme, or more extreme, than the sample maximum.

Consider a random sample of continuous random variables $X_1, X_2, ..., X_n$ with CDF $F$. Define the sample maximum: $$ M = \max(X_1, X_2, ..., X_n) $$ Some simulations I've done (see the R code ...
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1answer
19 views

Rigorous Order Statistics with Indicator Functions

If three people are randomly placed along a 1 mile road, the probability that no two of them are less than $m$ miles apart for $m \leq \frac{1}{2}$ could be solved by using the density for the order ...
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0answers
15 views

How do I describe probabilities of realised variables in terms of CDFs and PDFs?

I have a game theory problem here where the realisation of the random variable $Y₂$ (termed $y₂$ is observed in the first stage of the game. $Y₂$ is the second highest order statistic for $n-1$ ...
2
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1answer
56 views

Joint distribution of range $(R=X_n-X_1)$ and mid-range $(V=\frac{1}{2}(X_1+X_n)$order statistics

Let $X_1,X_2, · · · , X_n$ be independent and identically distributed Uniform random variables on the interval (0, a) for a > 0, each having a density function $f(x) = \frac{1}{a}$, $0<x<a$. Let ...
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1answer
16 views

Order statistics, no distribution, probability of percentile

$Let \ Y_1<...<Y_8 \\ be \ the \ order \ statistics \ of \ n \ independent \ observations\ from \ a \ continuous \ type \ distribution \ with \\ 70th \ percentile \ \pi_0._7 = 27.3$ The ...
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1answer
29 views

Find the Distribution of an Order Statistic

I am working on a practice prelim question. It states: Let $Y_1 < Y_2 < Y_3 < Y_4$ denote the order statistics of a random sample of size 4 from a distribution which is uniform on ...
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2answers
12 views

$Y_{(n)} = X_{(n)}/\mu$?

If $ X_1, ...,X_n$ are iid random variables such that $ X_i \sim U(0, \mu)$, is that true that if $Y_i = X_i/\mu$, then $Y_{(n)} = X_{(n)}/\mu?$ I am sorry if the question looks so simple and I am nt ...
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1answer
25 views

Given a set of partial orderings of samples from a set of distributions, can we estimate the (relative) mean of the distributions?

This problem is motivated by attempting to construct a total ordering out of an arbitrarily large set of potentially contradictory partial orderings. Let's assume we have some set of items I for ...
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0answers
28 views

probability that a variable, as a function of choice variables, is among the top k out of n when ordered

Suppose $(h_1,h_2,...,h_n)'$ is an $n\times 1$ vector. Let $h_i=g_iX_i$, where $g_i$ is a choice variable which can vary across $i$ and $X_i$ is a random shock with Pareto Type I distribution. ...
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0answers
31 views

Order statistics of a max-min problem.

$X_{A,1}, X_{B,1}, X_{A,2}, X_{B,2}$ are independent and identically distributed (i.i.d.) exponential random variables with parameter $\lambda=1$, i.e., the means of the four random variables are all ...
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1answer
45 views

Probability that a random variable is among the top k out of n when ordered

Suppose $X_1,X_2,\ldots,X_n $ are $n$ i.i.d. random variables with a continuous distribution $F(x)$ and density function $f(x)$. What is the probability distribution that any given $X_i$ is among the ...
3
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1answer
55 views

Expectation of first and second order statistics in a random distribution

Let $E(f_{i}^{n})$ and $E(s_{i}^{n})$ denote the expected first and second order statistics for $n$ draws from the distribution $V_i$ .i.e set $X_{i}^{n}=\{x^1,.....,x^n | x^j \sim V_i \}$ and let ...
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1answer
25 views

Testing a hypothesys about a survey?

A survey of $61, 647$ people including questions about office relationships. Of the respondents, $26$% reported that bosses scream at employees. Use a $.05$ significance level to test the claim that ...
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1answer
26 views

Proving some properties about the expected first order statistic (maximum) with respect to sample size.

Question: Consider $n$ random variables $x_1, x_2,\cdots x_n\sim \mathcal{N}(0,1)$. The expected value of the $i$th order statistic (the maximum) can be written as ...
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0answers
41 views

Threshold calculation

I need to calculate a threshold $[T]$ for a data set. The elements are a total count $[t]$ of items, and of those total items there will be variable counts of hot$[h]$ and Unknown$[U]$. So the ...
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30 views

Definite integral of product of regularized incomplete beta functions

I'm trying to find $$\int_{0}^{1} \! {I_x'(\alpha_1,\beta_1) I_x(\alpha_2,\beta_2) \dots I_x(\alpha_n,\beta_n)} \,\mathrm{d} x$$ where $\alpha_i, \beta_i$ are constant. Note the first term is ...
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0answers
19 views

Distribution maximum with small sample related to large sample

Suppose the random variables $X_i$, $i=1,\cdots,n$ and $Y_j$, $j=1,\cdots,m$ all have distribution $F(x)$, with order statistics denoted by $X_{(i)}$ and $Y_{(j)}$. Assuming $n<m$ (e.g. $n=m/100$), ...
3
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0answers
40 views

Gamma distribution Norming constant for extreme minima

the norming constants for extreme maxima of Gamma distribution is known and is give in link.springer.com/article/10.1007/s10687-010-0125-3. I would like to know is there reference or paper that states ...
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1answer
36 views

How to calculate $\int\limits_{-\infty}^{+\infty} \frac{n-1}{\alpha}\Phi(\frac{x}{\alpha})^{n-2}\phi(\frac{x}{\alpha})^2dx$?

I was working on a research project that involves taking the integral of $$\frac{n-1}{\alpha}\int\limits_{-\infty}^{+\infty} ...
0
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1answer
72 views

Normalizing constants for Extreme value distributions

I have a question regarding the normalizing constants $\mu$ and $\sigma$ that appear in the following problem. Let the random variable $Y_n$ be $Y_n=max(a_1,a_{2},\cdots, a_n)$ and $X_{n}$ be ...
1
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2answers
88 views

Formulation and computation of “the” unique median of an even-sized list

Consider an even-sized set of numbers $X = \{x_k\}$, such as $X = \{1, 2, 7, 10\}$. The median $m$ is defined as: $$m = \mathrm{arg \min_x} \sum_k \lvert x_k - x\rvert^1$$ Any $m \in [2, 7]$ is a ...
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1answer
37 views

Question about Logistic Regression - 7

I am currently studying Logistic Regression. I am facing a problem with understand the sentence in the red circle below. I am trying to figure out what he/she means by the sentence. Please let me have ...
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0answers
26 views

Question about Logistic Regression - 6

I am studying Logistic Regression and I have come across to understating the paragraph below. I kind of can understand, but it makes me confused when I read the sentence in the red circle, "It also ...
0
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1answer
32 views

Question about Logistic Regression - 2

How should I tell the difference between those two formulas in the circles below. I am studying logistic regression and I have faced two different formulas from two different documents. I don't know ...
1
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2answers
33 views

Finding the correlation coefficient of ordered statistics

I am working on the following problem. Let $$X_{(1)}, \ldots ,X_{(n)}$$ be the order statistics from the uniform distribution of $[0,1]$. Find the coefficient correlation of $X_{(1)}$ and ...
4
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3answers
53 views

Question about English sentences in statistics?

Can somebody help me interpreting the red circled sentences in planer English? I understand "We view $y_i$ as a realization of a random variable $Y_i$ that can take the values of one and zero" but ...
0
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2answers
52 views

convert continuous random variable to a discrete one for the given exponential distribution

I understand that the following question requires converting continuous r.v. to discrete r.v. But How can we get a PMF from the CDF of continuous distribution? It involves dividing continuous values ...
2
votes
3answers
253 views

In how many ways can 4 girls and 3 boys sit in a row such that just the girls are to sit next to each other? Answer: 288

In how many ways can 4 girls and 3 boys sit in a row such that just the girls are to sit next to each other? Answer: 288 Please explain how to get this. I understand that we have GGGG => 4 ...
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1answer
15 views

How do I compare how much variation there is between data sets?

I have a large number (~1000) of number sets containing 8 numerical elements. Here's an example: ...
3
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2answers
183 views

Variance of a function of independent random variables

Suppose I have two discrete independant random variables $X$ and $Y$, and that I'm interested in the expected value of the random variable $W$, where: $$ W= \text{sign}(X-Y). $$ So, W is 1 if ...
0
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0answers
23 views

RMSE to rank ordering error

Consider an iid sequence of standard normal random variables $ \lbrace X_i \rbrace$ , where $1 \leq i \leq n$. Consider that $n$ is large. Suppose we have a realization $\lbrace x_i \rbrace$, such ...
-1
votes
1answer
47 views

Distribution of minimum of independent normal variables

Suppose $X_t\sim N(\mu,\sigma^2t)$, $X_t$ are independent. Is the distribution of $$\min_{0\leq t\leq T}X_t$$known? In other words can this probability be found $$P(\min_{0\leq t\leq T}X_t\leq a)?$$ I ...
1
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2answers
123 views

Probability that order statistic is larger than the other

Given the density function: $$f_Y(y)=e^{-(y+1)}, y>-1$$ Let $Y_1,..,Y_4$ be a random sample from the distribution defined by the density function above. Let $Y_{(1)},..,Y_{(4)}$ be the ...
2
votes
1answer
118 views

Function to predict processing service overload

We have a black box that for each input request a, it outputs a computed response b. The computation time for a given request varies in a stable way over time. Stable means here that it is still ...
-1
votes
1answer
22 views

Poisson distribution- finding $\lambda$ with only $x$-values [closed]

Suppose that in a poisson distribution, it so happens that $\Pr(X=2)=\Pr(X=4)$. What is $\lambda$? What is $\Pr[X=x]$?
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1answer
35 views

Converting failure rates between periods

I'm trying to figure out how to convert an annual failure rate between periods. Assume failures are uniform and independent. I know that the quick, back-of-the-envelope way is simply to divide the ...
-1
votes
1answer
59 views

Proof of Expectation Formula

Prove that $E(X) = \mu$, where $X$ is the distribution of the sample mean and $\mu$ is the population mean. That is, the expected value of the sample mean $X$ is equivalent to the population mean. ...
1
vote
2answers
62 views

Maximum Likelihood Estimator for Uniform Distribution

Can somebody please explain this example to me. I am struggling to see why the likelihood is $\frac{1}{\theta^n}$ only if theta is greater than the maximum x. Furthermore, why is it the case that ...
1
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0answers
24 views

What conditions must satisfiy a positively skewed density function to ensure that median is greater than mean

We are collecting environmental Air Quality data. When we validate data, we always plot ECDF and compute basic statistics and percentiles. Our experimental distributions are far away from normality. ...
0
votes
0answers
28 views

Normal Distribution while finding sigma

I was reading some things about normal distribution and saw this problem in a text a couple days ago. I know it might be a little advanced for me at the moment, but I was wanted to know if someone can ...
0
votes
0answers
30 views

Inequality with CDF of order statistics

here is a problem I have been struggling with for a while now. This is for a paper I am working on. Any help would be appreciated! Here we go: Let $\theta _{i},$ $i=1,...,N$, be drawn independently ...
4
votes
1answer
2k views

Distribution of the sum of the $q$th largest observations to the sum of total for a power-law.

Where $X_{(1)}, X_{(2)}, \ldots,X_{(n)}$ are sorted independents r.v.s, where we index and order in such a way that $X_{(i)} \geq X_{(i-1)}$, $i>1$ where all realizations follow the same Standard ...
0
votes
2answers
33 views

Show probability is NOT 0.5 for independent events?

Let $Ω$ denote the sample space. Let $A ⊆ Ω$ such that the events $A$ and $B$ are statistically independent for all $B ⊆ Ω$. Show that $P(A) ≠ 0.5$. I have no idea how to go about this. I'm not ...
1
vote
0answers
42 views

Ancillary statistics and the use of Basu theorem

Let Y1, Y2, ....., Yn be the order statistics of a random sample of size n from the distribution exp(theta) If R= ( n Y1 / M) where M = summation Yi . To show that R and M are independent ?? ...