# Questions tagged [extreme-value-analysis]

This tag is for questions pertaining to the probabilistic/statistical theory of extreme deviations from the median of probability distributions. A central result of this theory is the Fisher–Tippett–Gnedenko theorem. It is not to be confused with the extreme-value-theorem tag that refers to a theorem for real valued continuous functions on a closed and bounded interval.

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### Fraction of the largest element of a sum of N i.i.d. random variates sampled from some distribution

I want to figure out how the fraction of the biggest sample of the sum of some iid samples from an arbitrary distribution varies. For example, I have a random variable $X \sim f(x)$, and I sampled $n$ ...
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### What are the extremes of the function $f\left(x,y\right)=xy\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}$?

$f\left(x,y\right)=xy\sqrt{1-\frac{x^2}{a^2}-\frac{y^2}{b^2}}$ ; where a,b are constants. I have tried to differentiate the equation, but I get stuck in a very hard system after this. The answer of ...
1 vote
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### Verification of Solution for Finding Minimum and Maximum of a Function in a Given Region

I want to find the minimum and maximum values of the function $f(x, y) = \ln(1 + 9x^2 + 4y^2)$ within the region $1 \leq \frac{x^2}{4} + \frac{y^2}{9} \leq 4$. My solution steps are as follows: ...
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### Determining maximum and minimum of the function

Determine the minimum and maximum of the function $f(x,y) = xy$ in the annulus $A = \{(x,y) \mid 1 \leq x^2 + y^2 \leq 4\}$. I'm wondering if in cases like this, where we have two conditions for the ...
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1 vote
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### Maximize $\surd(K_1)+2\surd(L_1)+3\surd(K_2)+4\surd(L_2)$ with constraints $L=L_1+L_2, K=K_1+K_2$

From my computations, I have found $$K_1^{*}=K/10, L_1^{*}=L/5, K_2^{*}=9K/10, L_2^{*}=4L/5, \lambda_1=\surd(5/2K), \lambda_2=\surd(5/L)$$ My Question is: How to find extreme value function $M(K, L)$. ...
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### Which claim is correct regarding the extrema of a function?

I have a function $f$ and I know that the function has only four unique extrema values which values are fA, fB, fC, and fD; moreover, I know that If condition A holds then fA is a local maximum of ...
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### Is there any hope to find extrema points of the given function $f_z (x,y)$ without using the Hessian method?

I have a three-variable function for $(x,y)\in[-\pi,\pi]$ and $z>0$ $f_z (x,y)=A(z)(\cos x+\cos y)+B(z)(\cos 2x+\cos 2y)+C(z) (\cos x \cdot\cos y),$ in which the real functions $A(z),B(z),C(z)$ can ...
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1 vote
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### $f(x, y, z) := xy + z^2$ on unit circle $B := \{(x, y, z) \in \mathbb{R}^3, x^2 + y^2 + z^2 \leq 1\}$

We look at $f(x, y, z) := xy + z^2$ on unit circle $B := \{(x, y, z) \in \mathbb{R}^3, x^2 + y^2 + z^2 \leq 1\}$. A) Explain why $f$ has a minimum and maximum on $B$. B) Show that $f$ has no local ...
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### Why the limiting distribution $M_n^*$ will be non-degenerate distribution?

Let $X_1,\dots, X_n$ be iid data with common distribution $F$ and $M_n:=\max\{X_1,\dots, X_n\}$. The distribution of $M_n$: $$P(M_n\le x)=F^n(x).$$ In the textbook, it says we proceed by looking at ...
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### Determine the extreme values of the function $f$ with $f(x,y)=x^2y(4-x-y)$

Determine the extreme values of the function $f$ with $f(x,y)=x^2y(4-x-y)$ for the points located inside and on the triangle delimited by the lines $x=0$, $y=0$ and $x+y=6$. Is this the function I'm ...
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### Why is a+b-c least here?

This part is from the well-known book What is Mathematics. Chapter VII (Maxima and Minima) - Section 5 (Steiner’s Problem) - Article 3 (A Complementary Problem). The problem: a, b, c is the distance ...
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### Extreme values of function with two variables $z=\frac{ax+by+c}{\sqrt{x^2+y^2+1}}$

I've been struggling with following problem: Determine extreme values of function $$z = \frac{ax+by+c}{\sqrt{x^2+y^2+1}},$$ where $a^2+b^2+c^2 \neq0$. Here is my approach: As we know differential ...
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### Find extreme values of ellipse

I have the curve equation $$Ax^2 + By^2 + Cxy = 1$$ which represents ellipse with center in (0, 0) and rotated some angle. How can I find max X and Y values (not semi-axes) on this ellipse? Points ...
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### Relative Extrema using Lagrange Multipliers - is it correct?

Here is the problem: Find an extrema of $f(x,y)=y^2+x$ with the given condition $x^2+2y^2=1$. I solved it this way: $L(x,y,\phi)=y^2+x+\lambda(x^2+2y^2-1)$ $\frac{\delta L}{\delta x}=1+2\lambda x =0$...
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### Points of $f(x)=x^3$ except the origin

I want to know what properties does the non-zero points in the domain of $f(x)=x^3$ have? For example, $0$ is an inflection point, since the curvature changes sign at $0$ ($f''$ changes sign near $0$)....
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### Let $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that $f''(x)\geq a$ for all $x \in \mathbb{R}$. Show that $f$ has an absolute minimum. [closed]

Let $f \colon \mathbb{R} \rightarrow \mathbb{R}$ twice differentiable such that $f''(x)\geq a$ for all $x \in \mathbb{R}$ and some $a>0$. Show that $f$ has an absolute minimum. I'm trying to ...
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### Showing that the distribution of record times $(\tau_k)_{k\geq 1}$ doesn't depend on the distribution, $F$, of the records $X_i$

I read that it's possible to show that the distribution of a record time sequence doesn't depend on the distribution of the record sequence itself, but how would one do this? So $(X_i)$ is an iid ...
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### Global extreme points of function wrt. two variables

Hi I am currently working on a problem in theoretical chemistry and am struggling a bit. While discussing Born-Oppenheimer energy functions, I was asked to find all the extreme points and saddle ...
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### Is $(0,0,0)$ saddle point?

$f(x,y,z)=x^3+y^3+z^3-3xy-3yz$ Is $(0,0,0)$ saddle point? $\nabla f(0,0,0)=0$, so $(0,0,0)$ is one of the stationary point. Also, because of my posture The reason why $f(0,0,0)$ is not a extreme value....
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### The reason why $f(0,0,0)$ is not a extreme value.
$f(x,y,z)=x^3+y^3+z^3-3xy-3yz$ According to wolfram alpha, $f(0,0,0)$ is not a extreme value. I considered why $f(0,0,0)$ is not a extreme value. Is this reason correct? When $x$ is very close to $0$ ...