This tag is for questions about order statistics and for questions in which one wants to calculate a probability, expected value, variance, standard deviation, or similar quantity of order statistics.

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3
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1answer
29 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 ...
0
votes
1answer
21 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 ...
0
votes
1answer
21 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 ...
0
votes
0answers
11 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 ...
0
votes
0answers
14 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 ...
0
votes
0answers
43 views

Total Integral of ordered joint probability is 1 ???

I have four random variables $X_1$, $X_2$, $X_3$ and $X_4$. Their joint dist. is $f(x_1,x_2,x_3,x_4)= \exp(-x_1-x_3)$, where limits are $x_4 = 0$ to $\infty$, $x_3 = x_4$ to $\infty$, $x_2 = x_3-x_4$ ...
0
votes
0answers
16 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
votes
0answers
28 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 ...
0
votes
1answer
30 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
votes
1answer
32 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
vote
2answers
74 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 ...
1
vote
1answer
31 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 ...
0
votes
0answers
24 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
votes
1answer
26 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
vote
2answers
29 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
votes
3answers
45 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
votes
2answers
26 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
43 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 ...
1
vote
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
votes
2answers
172 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
votes
0answers
22 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
46 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
vote
2answers
120 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
20 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 $\Pr[X=x]$?
0
votes
1answer
28 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
43 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
44 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
vote
0answers
22 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
24 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
25 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 ...
3
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
31 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
33 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 ?? ...
0
votes
2answers
70 views

Question on finding the MLE [closed]

I would appreciate any input or direction on this: I need to find the MLE of the hazard rate $\lambda$ where $$F(y;\lambda) = \begin{cases}1-e^{-\frac{y}{\lambda}} & y \ge 0, \\0 & y < ...
0
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0answers
53 views

Question about computing the sample mean and variance values from a sample coming from a Weibull Distribution …

Let's suppose that I have a random sample x from a Weibul distribution with shape parameter k=1 and scale parameter λ=2... How am I supposed to compute the mean value of the sample ? Also what can I ...
0
votes
1answer
45 views

How do I find $\theta$ with bootstrap?

I have two vectors of known values $x$ and $y$. And the relationship between them is $y=\sin(\theta \cdot x)+\epsilon$, $\epsilon \sim N(0,1) $ . The question is how do I estimate $\theta$ with ...
0
votes
1answer
37 views

Joint PDF of all n Order Statistics

If $X_1,\ldots,X_n$ is a random sample from a continuous distribution with pdf $f_{\theta}(x)$, why is the joint PDF of the order statistics $X_{(1)},\ldots,X_{(n)}$ the following: $$\large ...
2
votes
0answers
32 views

The distribution of the ith order statistic for discrete random variables

Assume $(X_i)_{i=1,...,n}$ are a sequence of real iid random variables with continuous density $p_x$. We know that $$Y:=\sum_{i=1}^n 1\{X_i\leq u\}\sim Bin(n,F_x(u)),$$ since $1\{X_i\leq u\}\sim ...
1
vote
0answers
52 views

Minimum of N Chi-square random variables when N is large

I have a problem in numerically evaluating the PDF of $Y=\min(X_1,X_2,\cdots,X_N)$ where $N=\binom{M}{K}$, the binomial coefficient and $X_i$s are iid Chi-square random variables. The CDF of $Y$ is ...
0
votes
0answers
33 views

Integral of a power function of a CDF

Is there a general formula to compute the integral $\int_0^s(F(x))^Ndx$ without knowing the exact expression of $F(x)$ but knowing that it is a CDF? What about $\int_0^sF(x)[1-F(x)]dx?$
0
votes
1answer
47 views

Difficult confidence interval problem

just needed some help on the following question. I've already attempted it twice but still can't get the answer out... Let $X_i,$ $i=1,...,n$ denote a random sample of size n from a population with ...
0
votes
1answer
31 views

Conditional PDF of an order statistic

Let $Y_1,\ldots,Y_n$ (all scalars) be random draws from a common CDF $F$ that has a PDF $f$. As usual, $Y_{(r)}$ denotes the $r$-th order statistic (e.g. $Y_{(1)}$ is the smallest). Let $z$ be fixed ...
0
votes
2answers
18 views

Area under the PDF of an order statitics

Consider a continuous random variable $$X=\min\{Y_1,Y_2,Y_3\}$$ where $Y_1,Y_2,Y_3$ are iid, non-negative random variables having the same PDF, $f_{Y}(x)$ and CDF $F_{Y}(x)$. The PDF of X is ...
1
vote
1answer
47 views

Find the limiting distribution of the following random variable

Let $X_1,X_2,...$ Be independent random variables with common density: $$f_X(x)=\alpha x^{-(\alpha+1)}. x>1$$ Where $\alpha>0$. Define a new sequence of random variables: ...
1
vote
1answer
23 views

$x1,x2~U(0,1)$,$Y$ is one of them who is closest to an end point. Find distribution of $Y$.

Let $X_1$ and $X_2$ be independent, $U (0, 1)$-distributed random variables, and let $Y$ denote the point that is closest to an endpoint. Determine the dis- tribution of $Y$. It's a question in ...
1
vote
1answer
25 views

Three part question

According to Labor Statistics, 75% of the women 25 through 49 years of age participate in the Labor force. Suppose 78% of the women in that age group are married. Suppose also that 61% of all women ...
1
vote
1answer
102 views

Stein's Method and Coupling of random variables

Suppose a particle starts at position 5 on a number line and at each period the particle moves one position to the right with probability p and, if the particle is above position 0, moves one position ...
0
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1answer
48 views

what is the difference between maximum likelihood estimation and usual probability inference? [closed]

Can somebody tell me a clear difference between MLE from the usual probability inferences?
1
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1answer
195 views

Order Statistics Example : Electronic components of a certain type…

Electronic components of a certain type have a length life (in hours) X, that follows the exponential distribution with probability density given by $$f(x) = \frac{1}{100}e^{-x/100}, x > 0 $$ a. ...