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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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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7 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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20 views

How to calculate $P(X_{(1)}< m < X_{(10)})$ there $m$ is the median. Have I done right?

I know that $$P(X_{(1)}< m < X_{(10)})=1-P(X_{(1)}>m)-P(X_{(10)}<m)$$ but is it right that we approach it by following? $$P(X_{(1)}< m < X_{(10)})=P(X_{(1)}<m \cup ...
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1answer
22 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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19 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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33 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
47 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 ...
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1answer
41 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 ...
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1answer
25 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 ...
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56 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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32 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
42 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 ...
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1answer
36 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 ...
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33 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
25 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 ...
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1answer
40 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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173 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 ...
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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. ...
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1answer
18 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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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 ...
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1answer
77 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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49 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 ...
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65 views

Find unbiased estimator for $\theta$ when PDF of X is $e^{\theta-X}$?

I know the MLE for $\theta$ is $min{[X_i]}$ but I can't check if that's unbiased because I don't know how to solve (U=$min{[X_i]}$ here) $\int_\theta^\infty u*f_U(u)du$ = $\int_\theta^\infty ...
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35 views

Find minimum variance & C.R.L.B?

Let $X_1$, $X_2$, $X_3$, ... , $X_n$ be a random sample from a distribution having mean µ and variance $σ^2$. Also, Let $Y_1$, $Y_2$, $Y_3$, ... , $Y_m$ be a second random sample from the same ...
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1answer
17 views

Variance measure for categorical data?

I have four discrete categories, each category has a sample count, and I'd like some measure of variance, where minimum variance is counts are evenly divided among all four categories and max variance ...
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44 views

The probability involving the ordered normal sample.

Let $X_1,X_2,X_3,X_4,X_5$ be a normal sample, taken from the distribution with unknown mean $\mu$ and known variance $\sigma^2$. Calculate $$P(|X_{3:5}-\mu|<0.841\cdot\sigma).$$ Comment. If the ...
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23 views

Determining the weights of known parameters in a formula

I have a formula of the following form: $a_1*w + a_2*x + a_3*y + a_4*z$ In the above formula, the $a_i$s can be thought of as weights to the corresponding parameters. The values of the ...
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18 views

What is asymptotic variance and how do I find it?

I don't understand the concept of asymptotic variance. Given the mle of a probability function, the likelihood function and the random variables how do I find asymptotic variance? What exactly is ...
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44 views

CDF of minimum of correlated and iid random variables

Consider two random variables $X_1=\min (W_1, W_2)$ and $ X_2=\min (W_3, W_4),$ where $W_1$, $W_2$,$W_3$ and $W_4$ are positive, identically distributed random variables. While $W_1$, $W_2$ are ...
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1answer
37 views

Variance of event counting

I have this question (not homework, review problem for qualifying exam), tried approaching it a couple of ways (unsuccessfully). Any recommendations? Let $X_1,..,X_n$ be i.i.d continuous rvs. A ...
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1answer
43 views

Probability distribution of $Y_1=min(X_i)$

Let $Y_1<Y_2<\ldots<Y_n$ denote the order statistics of a random sample of size $n$ from the distribution with pdf : $$f(x;\theta)=e^{-(x-\theta)}I_{(\theta,\infty)}(x)$$ Here we use the ...
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36 views

Obtain distribution of mid-range in uniform

I want to obtain distribution of mid-range, $(x_{(1)} + x_{(n)})/2$, of an uniform(a, b) random variable. One can use the following transformation. $M = \frac{X_{(1)} + X_{(n)}}{2}$ and $W = ...
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19 views

iid and correlated order statistics - A comparison

This is an extension of my previous question from the post iid and correlated order statistics a comparison Consider the two random variables $X_1$ and $X_2$ defined via non-negative random ...
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1answer
26 views

iid and correlated order statistics a comparison

Consider the two random variables $X_1$ and $X_2$ defined via i.i.d, non-negative random variables $Y_1$ and $Y_2$ as $ X_1=\min(Y_1,Y_2)\\$ $ X_2=\min(Y_1,Y_1)$ - (maximally correlated) The ...
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1answer
31 views

Expectation of maximum of Binomial RVs

Given an iid sample $X_{1}, \ldots, X_{n} \sim Bin(n, p)$ I'm trying to find $$E(X_{(n)})$$ that is the expectation of the sample maximum. Unfortunately I don't know where to start. It seems that ...
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1answer
37 views

how to create unbiased estimator in uniform distribution

$X_{1}, X_{2},...X_{n},n\geqslant 2, $ is a random sample from unif[$\theta -1, \theta +1$] Followed with the problem, I got T(X)=($X_{(1)}, X_{(n)} $) is sufficient but not complete, But I got ...
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51 views

Expected value of power of exponential order statistics

Finding the expected value of the $\gamma$:th power of the $k$:th standard exponential order statistic in a sample of $n$ implies evaluating the integral: ...
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1answer
47 views

PDF of the r-th Order Statistic (essentially need help with a derivative or combinatorics)

So I get why it is that the CDF of the $r$th order statistic is $$ F_{X_{(r)}}(x) = \sum_{i=r}^{n} \binom{n}{i}[F_{X}(x)]^{i}[1-F_{X}(x)]^{n-i} $$ but I'm not seeing how to prove that the PDF is ...
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1answer
58 views

Xmas Special: 25 identical sweets shared between two indivuduals and…

Ready?.. 25 identical sweets must be shared bewteen 1 boy & 1 girl. Each of the children MUST recieve at least 10 sweets each. all sweets must be distrubuted. order not important although I will ...
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2answers
36 views

Statistics: Odd Moments

Need help with this stat question. I know you start by integrating $z^k f(z)$ from $-\infty$ to $0$ + integral of $z^k f(z)$ from $0$ to $\infty$. After that I'm stuck.
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22 views

Evenly selected order statistic?

Recently, I'm studying Nonparametric methods, especially Order Statistics. And I saw a sentence which says that Order statistic has to be evenly selected. And now I'm wondering what 'evenly ...
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41 views

probability that at least one observation of a random sample

What is the probability that at least one observation of a random sample of size n=5 from a continuous type distribution exceeds the 90th percentile. Let W~Binomial n=5, P=?. how can i get the p of ...
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1answer
24 views

Bayesian Statistics: Estimators and Posterior Probability

If I let $M ∼ Γ(α,β)$ (where $α, β$ are known) Let $X_1,...,X_n$ be discrete random variables such that $X_i$|$θ$ ∼ i.i.d. Poisson with parameter $θ$, where $θ$ is a realization of $M$. I have two ...
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17 views

Use condition to make r.v.s independent and then use Laplace transform to sum them up?

Let $X_1,X_2$ be two nonnegative i.i.d. random variables. Let $k_1 = \text{argmax} (X_1, X_2)$. This means, $k_1$ is also a random variable. Now I define $c_k$ as $c_k = A$ if $k = k_1$, and $c_k = ...
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1answer
31 views

Difference of two largest values from a set of i.i.d. random variables

I have a set of i.i.d. random variables $(E_1, \ldots, E_N)$ that are exponentially distributed with parameter $\lambda$. I guess, if $E_1$ lives in $(S,\mathcal{S},\mathbb{P})$, then the whole set ...
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33 views

Order statistics of mixed (iid as well as non iid) random variables

Does anyone know if there are results (PDF or the CDF) on the order statistics (at least minimum or maximum) of $n$ random variables in which a few of them are i.i.d. and the rest of them are ...
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96 views

Conditional expectation to de maximum $E(X_1\mid X_{(n)})$

Let $X_1, \ldots, X_n$ a random sample of a Uniform(0,1): Which is $E(X_1\mid X_{(n)})$ ? where $X_{(n)}=\max\{X_1,\ldots,X_n\}$
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1answer
54 views

Joint distribution $(X_1,X_{(n)})$ order statistics

Let $X_1, \ldots, X_n$ a random sample of a Uniform(0,1), I want to show which the joint distribution of $(X_1,X_{(n)})$ is. I do the following: $$ P(X_1\leq x, X_{(n)}\leq y)=P(X_1\leq x, X_1\leq y, ...
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58 views

Question about piecewise exponential distribution

This is an excerpt from the following paper. I am particularly interested in knowing how the authors got the displayed equations. We let [Z] denote the distribution of a generic random variable ...
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1answer
56 views

Difficulty to compute an integral

Have somebody ideas to evaluate the following integral ? $$J_n=\int_{-\infty}^{+\infty} \left(\frac{\pi^2}{4}-\arctan(x)^2\right)^n\,dx$$ I'm trying this because I have shown that the empiric ...