All Questions
Tagged with extreme-value probability
77 questions
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Expectation of the minimum of random variables (Exponential + Erlang)
Consider the following random variable
$$
Z=\min_i\{X_i+Y_i\}
$$
for $-n\leq i\leq n$, where $X_i\overset{\mathrm{iid}}{\sim}\text{Exp}(\lambda)$, $Y_i\overset{\mathrm{iid}}{\sim}\text{Erlang}(|i|,\...
- 150
3
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1
answer
85
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Maximum of two independent gamma variables
Let $X_1$, $X_2$ be two independent random variables with different gamma distributions, and $X = \max\{X_1, X_2\}$.
Are there known results for the distribution of $X$? Actually I only need to know $\...
- 1,191
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0
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51
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How to understand intuitively the CDF formula for the maximum statistic of three iid rv’s? [duplicate]
Given that all three iid rv’s ($X_1, X_2, X_3$) have CDF $F(x)$, the formula for the CDF $G(y)$ of the largest rv ($Y=X_i$) among the three is:
$G(y)=P(X_1 \leq y) \cdot P(X_2 \leq y) \cdot P(X_3 \leq ...
2
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72
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$1-F$ is rapidly varying if and only if there exists $b_n$ such that $\frac{\max X_i}{b_n} \to 1$ in probability
The following is a problem from Extreme Values, Regular Variation and Point Processes by Resnick.
We will say $1-F$ is rapidly varying as $x \to \infty$ if $\lim_{t \to \infty} \frac{1-F(tx)}{1-F(t)} =...
- 656
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42
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Is modeling the extreme value of a distribution a basic probability result?
I was reading briefly about the field of EVT - extreme value theory, and the associated distributions that arise from modeling the maximum of a finite sample. It's not quite clear to me the nature of ...
- 64.8k
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2
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221
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Calculating probability related to maximum of random variables
Let $X_1, X_2, \cdots, X_n$ be non-negative continuous iid random variables. The goal is to find the probability:
\begin{align*}
\Pr(\max_{k+1 \leq i \leq j } X_i < \max_{1 \leq i \leq k }X_i)
\end{...
2
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1
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178
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CDF of max of $n$ cauchy variates
Suppose $X_1,X_2,\cdots,X_n$ are iid copies of a standard cauchy variate with pdf
$$ f(x)=\frac{1}{\pi(1+x^2)},0<x< \infty. $$
Define:
$$ Y=1+ \underset{1 \leq i \leq n}\max X_i.$$ I want to ...
- 447
2
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1
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398
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Why does Gumbel distribution have two different expressions?
Let $X_1,X_2,\dots,X_n$ be iid random variables with distribution function $F(x)$ and $M_n:=\max\{X_1,\dots,X_n\}$. By the extreme value theorem, there exist two sequences of real numbers $a_n>0$ ...
- 747
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1k
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Distribution that doesn't belong to any maximum domain of attraction?
Question
Does there exist a (non-degenerate) distribution that does NOT belong to any maximum domain of attraction?
That is:
Does there exist any non-degenerate probability distribution function $F$ ...
- 223
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0
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39
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Does independence implies independence conditionally on max of the data?
Let be $X_1, ..., X_n$ I.I.D. numerical random variables with contiunous density $f$.
Note $M(X) = \max(X_1, ..., X_n)$ their maximum.
Are $X_1, ..., X_n$ independent conditionally on $M(X) = x$ for ...
- 2,629
1
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1
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70
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Numerical superiority necessary to beat in $L^\infty$ a population one standard deviation ahead
Suppose $m$ independent random variables $X_i$ have the distribution $\mathcal{N}(0, 1)$, and $n$ independent random variables $Y_j$ (also independent of the $X_i$) have the distribution $\mathcal{N}(...
- 11
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Given 5 variables, all independently normally distributed, what is the probaility that variable A is lower than the other 4 variables?
Suppose variables A B C D and E are independent, normally distributed, with known variance and mean.
What is the probability that A is less than B and C and D and E?
Essentially, I have model ...
- 11
4
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319
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computing $P\left(\max(U_{(1)}, U_{(2)}-U_{(1)}, \cdots,U_{(n)}-U_{(n-1)} ) <a\right)$
Let $U_{1}, \, ... \, ,U_{n}$ be a random sample of uniform random variables $U_i \sim \mathrm{Uniform}(0,1)$. Let $U_{(1)}, \, ... \, , U_{(n)}$ be the order statistics of the sample. My problem is ...
4
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1
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453
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Residuals in Generalized Pareto Distribution
I'm learning generalized Pareto distribution for fitting extreme value data. I came across an R package evir that is able to plot residuals. Residuals from a GPD ...
- 8,655
1
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0
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229
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Gutenberg-Richter Recurrence Law: why is rate defined as probability of being exceeded?
According to Kramer (1996): Guttenberg and Richter gathered data for Southern California earthquakes over a period of many years and organized data according to the number of earthquakes that exceeded ...
- 463
2
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1
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693
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The probability that the minimum of a multivariate Gaussian exceeds zero
Suppose $$X \sim \mathcal N_n(\text{diag}(\Sigma), \sigma^2 \Sigma)$$
where $\Sigma$ may be allowed to be low rank, and let $Y = \min_i
> X_i$.
What can be said about $P\left(Y \geq 0\right)$?
In ...
- 20.8k
1
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0
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142
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Expectation of a sequence of random variables based on a set of iid Gaussian random variables
This is a rather convoluted problem: I'll my best trying to explain it. So, we have $m$ iid standard Gaussian RVs $Q_i$. We get a realization from each of them, and these values $q_1,\dots,q_m$ are ...
- 18.4k
3
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3
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1k
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Expectation of the minimum of a continuous random variable $X$ and a discrete random variable $Y$
Let $X\sim Exp(1)$ and independently let $Y$ have the pmf $P(Y=k)= p$, $P(Y = \infty) = 1-p$, where $k < \infty$. I'd like to calculate $\mathbb{E}(Z)$, where $Z = \min(X,Y)$.
Usually, we tackle ...
- 309
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43
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Question about GEV
I'm doing some analysis involving rectangular pulse processes.
Suppose for each process {Xi} that X changes after equal so-...
- 413
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0
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491
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Obtaining the probability of exceedance corresponding a given return period
I have a time series of data (15 years). Following plots show the fitted PDF (generalized extreme value distribution) and corresponding CDF (i.e. 1 minus CDF).
The data used here is not the total ...
8
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1
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409
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Are min$(X_1,\ldots,X_n)$ and min$(X_1Y_1,\ldots,X_nY_n)$ independent for $n$ to infinity?
Assume that we have given two continuous iid random variables $X$ and $Y$ with support $[1,c)$, where $c$ is some constant greater than one.
Now assume I have a given iid sample $X_1, \ldots,X_n$ and $...
- 81
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0
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48
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Exponential Inequality For Probability of Being Close to Maximum
Given $n$ independent identically distributed random variables $X_1, X_2, \ldots, X_n$ that have $|X_i| < \lambda$ for all $i$. Let $\max(X)$ be the maximum of these $n$ variables.
Is there a ...
- 103
1
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0
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19
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Probability of random population value being higher than sample maximum
Considering a small sample size (n < 10) from a population, I'm trying to find how likely a random population value would be greater than the maximum of the sample.
Hoping ye could help me with ...
- 11
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1
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52
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Two datasets with same length give different number of extremes
I have two datasets of a given variable x that have the same length, let's say 14600 values in total each one.
I need to extract the extreme observations within ...
- 344
7
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243
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Is there a random variable $X$ with positive support such that the ratio of the two smallest realizations of an iid sample goes to one?
Imagine I have given a random variable $X$ with supp$(X)=(0,\infty)$ and $\mathbb P(X \in (0,a))>0$ for any fixed $a>0$
Now given an iid sample $X_1,...,X_n$ - is it possible that
$$X^{(2)}/...
user244467
0
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0
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51
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Estimate true mean of the maximum of N sample means
Let's say we have N distributions $\mathcal N(\mu_i, \sigma_i)$, each with unknown mean $\mu_i$ and unknown standard deviation $\sigma_i$, $i=0,...,N-1$.
For each $i$, $M$ independent random samples ...
- 1
5
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1
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122
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Identity on expectation of the minimum of two iid random variables with bounded support
I am reading the 2008 annals of statistics paper "Ranking and empirical minimisation of U-statistics" by Clémençon et. al, and read a statement which I do not know why is true. In order to accurately ...
- 657
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0
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75
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Minimum of n independent, but not identically distributed inverse Gaussians
I would like to find the probability distribution of the minimum of of n independent, but not identically distributed, i.e. differently parametrized inverse Gaussians. I would prefer an analytical ...
- 13
3
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1
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1k
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Mean and variance of maximum of normal random variables
I'm trying to find the mean and variance of $Y = \max(X_1, ..., X_n)$ where $X_i \sim \mathcal{N}(\mu_i, \sigma^2)$.
Note that the $X_i$ are independent, but not identically distributed. That is, ...
- 301
1
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1
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174
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Cumulative Probability Distribution of Maximum and 2nd from Maximum of 4 Variables
I understand that the cumulative probability distribution cum(x) of the maximum of 2 variables x1 and x2 with probability distribution p1(x1) and p2(x2) is the product of the two cumulative ...
- 389
2
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2
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89
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$\mathbb{E}[\min (X_{1:n}) + \max(X_{1:n})]/2 = \mathbb{E}[\text{median}(X_{1:n})]$?
Say $X$ is continuous random variable, and we have $n$ iid samples, denoted as $X_{1:n}$. Then can we say the following
$$\mathbb{E}[\min (X_{1:n}) + \max(X_{1:n})]/2 = \mathbb{E}[\mathrm{median}(X_{...
- 1,565
3
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1
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142
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Distribution of distance of N-1 gamma distributed iid random variables from minimum
I have the minimum value of N iid random variables that are gamma-distributed. The parameters of the gamma distribution are known. What would be the distribution of the distance of the remaining N - 1 ...
- 1,759
1
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0
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364
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Normalising constant of the Gumbel in extreme value theory
Well known facts in extreme value theory:
Let $\{X_i\}_{\forall i \in \{1,...,n\}}$ be i.i.d. random variables with cdf $F$. If there exists $\{a_n\}_{n\in \mathbb{N}}>0$, and $\{b_n\}_{n\in \...
- 935
1
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0
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33
views
Compare maxima of two Bernouilli experiments
I am looking at the following question -- which has already been solved for the case of Gaussian samples Compare maxima of two Gaussian samples but I am unable to find a similar answer for the ...
- 11
2
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2
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1k
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Distribution of extreme values, case of uniform
Question: For $U_1 , \dots, U_n$ i.i.d. $U \sim \mathrm{unif}[0,1]$, we want to find the asymptotic distribution of $Z_n = n(1-U_{(n)})$ where $U_{(n)} = \max(U_1 , ... , U_n)$
I found this: ...
- 770
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1
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351
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Approximation/bound to a_n and b_n in normal maxima to Gumbel
I was reading this which tries to find $a_n$ and $b_n$ such that
$$F\left(a_n x+b_n\right)^n\rightarrow^{n\rightarrow\infty} G(x) = e^{-\exp(-x)},$$
where $F$ is the cdf of a standard normal.
The ...
- 252
3
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3
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741
views
Finding the mean of the max order statistic drawn from standard normal
Let $X=\max\{X_1, X_2, \cdots, X_N\}$, where each $X_i \sim N(0,1)$ and are independent. What is the approximate value of $X$ for large $N$.
The term "approximate" isn't defined very clearly. I'm ...
- 61
1
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0
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44
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Relationship between expected minimum as sample size increased
Suppose I have $N$ random variables $X_i$, $i = 1, \dots, N$. I am interested in the quantity
$$ A = \mathbb E \left[\min_{i=1, \dots, N} X_i \right]. $$
Now suppose I take a subset $S \subset \{1, \...
- 323
1
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1
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94
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Simple probability question (similar to birthday paradox)
If $x$ objects are randomly distributed to $n$ groups, what is the formula for working out how big $x$ needs to be for the probability that at least one of the groups gets an amount $y$ (or larger) to ...
- 41
5
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98
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Unbiased estimator for top-k bernoullis
Supposed I have $n$ coins and I'm interested in finding the $k < n$ coins which have the highest odds of coming up heads and I want to know $p(heads)$ for each of these $k$ coins.
Assume that I'm ...
- 241
3
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0
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85
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An arithmetic mean preserves normal distributions, maximum preserves Frechet/Gumbel/Extreme Value distributions, but what about all other power means?
Let the $k$-power mean of two numbers $x$ and $y$ be defined as $M^k(x,y) = \left(\frac{x^k+y^k}{2}\right)^{1/k}$.
For the case $k=1$, we have that if $X,Y$ are independently normally distributed, ...
- 1,594
1
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1
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225
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What is the probability of one peak over threshold (POT) event in a given year?
I have got a dataset of daily rainfall data (mm) from different hydrological gauges. The observed time period is not fixed, and therefore the gauges' time series have different lengths (i.e. 10, 23, ...
- 344
2
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0
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87
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When is variance of sample maximum greater than unconditional variance?
Let $X_1$,...,$X_n$ be $n$ i.i.d. RVs with continuous distribution $F$. Further let $X_{(1)}$,...,$X_{(n)}$ be the associated order statistics such that $X_{(1)}<X_{(2)}<...<X_{(n)}$.
Under ...
5
votes
1
answer
490
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Predict probability of rare event
Let's say I have a dataset about passages of cars on a road.
The dataset contains information about time, driver, car, weather, and most importantly whether the car was involved in an accident.
Of ...
- 103
4
votes
2
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121
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Extreme value distribution with unknown variance
Let $\{X_1,\ldots,X_n\}$ be a sequence of r.v. such that $X_i\sim N(0,\sigma^2)$.
It is usually stated in Extreme Value Theory textbooks that (for suitably chosen $a_n$ and $b_n$)
$$\mathbb{P}\left(\...
- 1,385
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1
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96
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How many random samples before you're not surprised by an extreme value?
Suppose I look at a collection of 10 students and calculate the mean and standard deviation of their GPAs. No one in this group has a 4.0. Then I take another 10 students at random and find that one ...
- 101
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1
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191
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How to find the maximum likelihood of this scenario occurring?
I have the equation: $ 0= 2.01106 - 0.00274(34.647+24.24a)-0.02059(45.647+21.122b)+1.37984(2.05-0.206c)-0.01176(10.588+11.963d)+0.00394(118.29-21.097e)-0.03552(92.17+2.855f)$
I have the above ...
- 103
3
votes
1
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2k
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Maximum of minimum of random variables
Is there a general formula for calculating distribution of the maximum of the minimum of random variables?
For example: say I have independent random variables $X_1,X_2,X_3$ distributed with CDFs $...
- 343
2
votes
2
answers
212
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P(X<Y|Z=t) where Z=min(X,Y)
Lets X and Y be uniform random variable where $x \in [0,a]$ and $y \in [0,b]$ where a < b. We design $Z=\min(X,Y)$.
I know that the CDF of Z is $P(Z<z)=1-\frac{(a-z)(b-z)}{ab}$
And by ...
- 719
1
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2
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370
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Why is $P(\min\{X_1,...,X_n\} ≥ y)=P(X_1≥y,..., X_n≥y)$?
Why is $$P(\min\{X_1,...,X_n\} ≥ y)=P(X_1≥y,..., X_n≥y)$$
and similarly
$$P(\max\{X_1,...,X_n\}≤y)=P(X_1≤y, ..., X_n≤y)$$
I.e. why are $\min$ and $\max$ equivalent to AND (since $P(X_1≥y, X_2≥y)$ ...
- 4,129