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Questions tagged [means]

In probability and statistics, mean and expected value are used synonymously to refer to one measure of the central tendency either of a probability distribution or of the random variable characterized by that distribution. For a data set, refers to a central value of a discrete set of numbers: specifically, the sum of the values divided by the number of values.

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Proofs of AM-GM inequality

The arithmetic - geometric mean inequality states that $$\frac{x_1+ \ldots + x_n}{n} \geq \sqrt[n]{x_1 \cdots x_n}$$ I'm looking for some original proofs of this inequality. I can find the usual ...
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7answers
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Do there exist pairs of distinct real numbers whose arithmetic, geometric and harmonic means are all integers?

I self-realized an interesting property today that all numbers $(a,b)$ belonging to the infinite set $$\{(a,b): a=(2l+1)^2, b=(2k+1)^2;\ l,k \in N;\ l,k\geq1\}$$ have their AM and GM both integers. ...
38
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5answers
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Prove inequality $\arccos \left( \frac{\sin 1-\sin x}{1-x} \right) \leq \sqrt{\frac{1+x+x^2}{3}}$

I was trying to figure out if the following function can serve as a mean (see mean value theorem): $$\arccos \left( \frac{\sin y-\sin x}{y-x} \right)$$ And turns out that for $x,y \leq \pi$ it does ...
36
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3answers
630 views

Repeatedly taking mean values of non-empty subsets of a set: $2,\,3,\,5,\,15,\,875,\,…$

Consider the following iterative process. We start with a 2-element set $S_0=\{0,1\}$. At $n^{\text{th}}$ step $(n\ge1)$ we take all non-empty subsets of $S_{n-1}$, then for each subset compute the ...
30
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3answers
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Geometric mean of reals between 0 and 1

What is the geometric mean of all reals between $0$ and $1$? I was thinking over this, but could not come up with anything useful. Please help me out.
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2answers
510 views

Finding properties of operation defined by $x⊕y=\frac{1}{\frac{1}{x}+\frac{1}{y}}$? (“Reciprocal addition” common for parallel resistors)

I have recently found some interesting properties of the function/operation: $x⊕y = \frac{1}{\frac{1}{x}+\frac{1}{y}} = \frac{xy}{x+y}$ where $x,y\ne0$. and similarly, its inverse operation: $x⊖y = ...
18
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1answer
496 views

A generalization of arithmetic and geometric means using elementary symmetric polynomials

Let $a_1, a_2, \ldots, a_n$ be positive real numbers. A while ago I noticed that if you form the polynomial $$ P(x) = (x - a_1)(x-a_2) \cdots (x-a_n) $$ then: The arithmetic mean of $a_1, \ldots, ...
18
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2answers
285 views

Which power means are constructible?

The three classic Pythagorean means $A$, $G$, $H$ (arithmetic, geometric, and harmonic mean respectively) of positive real $a$ and $b$ have a cute geometric construction, as does the quadratic mean $Q$...
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2answers
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What happens if you repeatedly take the arithmetic mean and geometric mean?

Given two positive real numbers, $A$ and $B$, such that $A\leq B$, take the geometric mean, giving $A'$, and the arithmetic mean, giving $B'$. Repeat ad infinitum. My intuition tells me that, since ...
17
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1answer
255 views

Integral results in difference of means $\pi(\frac{a+b}{2} - \sqrt{ab})$

$$\int_a^b \left\{ \left(1-\frac{a}{r}\right)\left(\frac{b}{r}-1\right)\right\}^{1/2}dr = \pi\left(\frac{a+b}{2} - \sqrt{ab}\right)$$ What an interesting integral! What strikes me is that the result ...
16
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3answers
261 views

Prove this integral $\int_0^\infty \frac{dx}{\sqrt{x^4+a^4}+\sqrt{x^4+b^4}}=\frac{\Gamma(1/4)^2 }{6 \sqrt{\pi}} \frac{a^3-b^3}{a^4-b^4}$

Turns out this integral has a very nice closed form: $$\int_0^\infty \frac{dx}{\sqrt{x^4+a^4}+\sqrt{x^4+b^4}}=\frac{\Gamma(1/4)^2 }{6 \sqrt{\pi}} \frac{a^3-b^3}{a^4-b^4}$$ I found it with ...
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3answers
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About an inequality including arithmetic mean, geometric mean and harmonic mean

For any $n$ positive real numbers $a_i\ (i=1,2,\cdots,n)$, let us define $A,G,H$ as $$A=\frac{\sum_{i=1}^{n}a_i}{n},\ G=\sqrt[n]{\prod_{i=1}^{n}a_i},\ H=\frac{n}{\sum_{i=1}^{n}\frac{1}{a_i}}.$$ Then,...
15
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2answers
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Mean vs. Median: When to Use?

I know the difference between the mean and the median. The mean of a set of numbers is the sum of all the numbers divided by the cardinality. The median of a set of numbers is the middle number, ...
14
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4answers
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Is a definite integral just a summation?

I am learning about definite integrals and found the formula for finding an average of a function over a given interval: $$\frac{1}{b-a} \int_{a}^{b} f\left(x\right) dx$$ If we look at the average ...
14
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3answers
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Understanding The Math Behind Elchanan Mossel’s Dice Paradox

So earlier today I came across Elchanan Mossel's Dice Paradox, and I am having some trouble understanding the solution. The question is as follows: You throw a fair six-sided die until you get 6. ...
14
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1answer
399 views

Is there a name for this “mean”?

We all know these means: $$GM = \sqrt[3]{xyz} $$ $$AM = \frac{x + y + z}{3}$$ $$QM = \sqrt{\frac{x^2 + y^2 + z^2}{3}} $$ Of course: $$GM \le AM \le QM $$ What about this one: $$XM = \sqrt{\frac{...
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9answers
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Providing that: $\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$

Let $a$ and $b$ be positive reals. Show that $$\lim\limits_{n\to\infty} \left(\frac{a^{\frac{1}{n}}+b^{\frac{1}{n}}}{2}\right)^n =\sqrt{ab}$$
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5answers
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Arithmetic mean. Why does it work?

I've been using the formula for the arithmetic mean all my life, but I'm not sure why it works. My current intuition is this one: The arithmetic mean is a number that when multiplied by the number ...
11
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1answer
678 views

Why is the $0$th power mean defined to be the geometric mean?

Mentioned in the wikipedia article, the $0$th power mean is defined to be the geometric mean. Why is this? I understand that a convenient consequence is that the means are ordered by their exponent. ...
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4answers
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Prove QM-AM inequality

$$\dfrac{x_1^2+ x_2^2 + \cdots + x_n^2}n \geq \left(\dfrac{x_1+x_2+\cdots+x_n}n\right)^2$$ I don't think AM, GM can be used here. And simple expansion doesn't help too. What should I do?
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4answers
835 views

A continuous function defined on an interval can have a mean value. What about a median?

A function can have an average value $$\frac{1}{b-a}\int_{a}^{b} f(x)dx$$ Can a continuous function have a median? How would that be computed?
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3answers
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Which number was removed from the first $n$ naturals?

A number is removed from the set of integers from $1$ to $n$. Now, the average of remaining numbers turns out to be $40.75$. Which integer was removed? By some brute force, I got $61$. I want to know ...
9
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5answers
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Proving the AM-GM inequality for 2 numbers $\sqrt{xy}\le\frac{x+y}2$

I am having trouble with this problem from my latest homework. Prove the arithmetic-geometric mean inequality. That is, for two positive real numbers $x,y$, we have $$ \sqrt{xy}≤ \frac{x+y}{2} .$$ ...
9
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1answer
227 views

Closed form for the integral $\int_0^\infty \frac{dx}{\sqrt{(x+a)(x+b)(x+c)(x+d)}}$

Let's consider the function defined by the integral: $$R(a,b,c,d)=\int_0^\infty \frac{dx}{\sqrt{(x+a)(x+b)(x+c)(x+d)}}$$ I'm interested in the case $a,b,c,d \in \mathbb{R}^+$. Obviously, the ...
9
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1answer
264 views

What is the average?

When I was first introduced to the concept of average (mean), I was confused. What does average mean? How does one number $\sum_{i=1}^{n} a_{i}$ represent the "central tendency" of a set of data ...
9
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2answers
198 views

Iterated means $a_{n+1}=\sqrt{a_n \frac{b_n+c_n}{2}}$, $b_{n+1}$ and $c_{n+1}$ similar, closed form for general initial conditions?

For every nonnegative $(a_0,b_0,c_0)$, consider $$a_{n+1}=\sqrt{a_n \frac{b_n+c_n}{2}},\quad b_{n+1}=\sqrt{b_n \frac{c_n+a_n}{2}},\quad c_{n+1}=\sqrt{c_n \frac{a_n+b_n}{2}}$$ $$M(a_0,b_0,c_0)=\lim_{...
8
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5answers
266 views

Prove that $\lim_{k\to\infty}{ \sqrt[n]{ \prod_{i=1}^{n}{(x_i+k)}} - k } = \frac{\sum_{i=1}^{n}{x_i}}{n}$

I accidentally discovered this equality, in which I can prove numerically using python. $$\lim_{k\to\infty}{ \sqrt[n]{ \prod_{i=1}^{n}{(x_i+k)}} - k } = \frac{\sum_{i=1}^{n}{x_i}}{n}$$ But I need ...
8
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1answer
90 views

Applied math: analyse cell orientations

I'm analyzing the orientation of cells and I stumbled over a peculiarity when I try to make a statement about the main direction of the cells and how many cells are oriented along this main direction. ...
8
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1answer
138 views

Prove that the $p$-mean is an increasing function of $p$

Let $p\neq0$ and $j=1,2,\cdots,n$ and $x_j>0$ and $$\chi(p)=\left(\frac{1}{n}\sum_{j=1}^nx_j^p\right)^\frac{1}{p}.$$ Prove that $\chi$ is strictly increasing and the following statements hold $\...
8
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2answers
132 views

Bounds for generalized mean $\frac{1}{2 \mu(a,b)}=\int_0^\infty \frac{dx}{e^{ax}+e^{bx}}$

Let us define a generalized mean of two positive real numbers $\mu(a,b)$ as: $$\frac{1}{2 \mu(a,b)}=\int_0^\infty \frac{dx}{e^{ax}+e^{bx}}=\int_0^1 \frac{dt}{t^{1-a}+t^{1-b}}$$ This integral has a ...
8
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0answers
239 views

Modified AGM: $a_{n+1}=\frac{a_n+b_n}{2}, \quad b_{n+1}=a_n+b_n-\sqrt{a_n b_n}$

The idea is as follows: generate a sequence from two numbers by subtracting their means from their sum, for example arithmetic and geometric means: $$a_{n+1}=a_n+b_n-\frac{a_n+b_n}{2}=\frac{a_n+...
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3answers
581 views

What is the maximum value of $\frac{x^{100}}{1+x+x^2+\ldots+x^{200}}$?

If $x$ is positive, what is the maximum value of this expression: $$\frac{x^{100}}{1+x+x^2+\ldots+x^{200}}$$ This question is from a book of problems on sequence and series under the section on AM-...
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3answers
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Derivation of mean and variance of Hypergeometric Distribution

I need clarified and detailed derivation of mean and variance of a hyper-geometric distribution. If a box contains $N$ balls, $a$ of them are black and $N-a$ are white, and $n$ number of balls are ...
7
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3answers
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How to show $\lim_{n \to \infty} \sqrt[n]{a^n+b^n}=\max \{a,b\}$? [duplicate]

Let $a\geq 0$ and $ b\geq 0$. Prove that $\lim_{n \to \infty} \sqrt[n]{a^n+b^n}=\max \{a,b\}$. [Hint: Use the identity $(a^n -b^n)=(a-b)(\sum_{i=0}^{n-1}a^ib^{n-1-i})$] I need some help! I cannot ...
7
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1answer
113 views

Median is twice the mean

I am stuck at solving the following problem (at what I believe is the last step): Determine which distributions on the non-negative reals, if any, with mean $\mu$ are such that $2\mu$ is a median. ...
7
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1answer
2k views

Importance of relation between harmonic, geometric,arithmetic and quadratic mean

Going trough my math textbook, I stumbled upon a proof of inequality of number means i.e. $$ H_n \le G_n \le A_n \le Q_n $$ where $H_n, G_n , A_n , Q_n$ are harmonic, geometric, arithmetic and ...
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2answers
174 views

Limit of a sequenced defined by arithmetic mean and geometric mean.

If $a_1 =\alpha$, $ a_2 = \beta$, and, for every $k$, $$a_{2k+1}= \frac{1} {2k} \sum_{i=1}^{2k} a_i\qquad a_{2k+2}=\left(\prod_{i=1}^{2k+1} a_i\right)^{1/(2k+1)}$$ what is the limit of the sequence ...
7
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1answer
191 views

Computing cube roots using three number means

I've asked a question some time ago about Computing square roots with arithmetic-harmonic mean but it turned out that this method is exactly the same as Newton's method (or Babylonian method) for ...
7
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2answers
297 views

A known closed form for Borchardt mean (generalization of AGM) - why doesn't it work?

There is a curious four parameter iteration introduced by Borchardt: $$a_{n+1}=\frac{a_n+b_n+c_n+d_n}{4} \\ b_{n+1}=\frac{\sqrt{a_n b_n}+\sqrt{c_n d_n}}{2} \\ c_{n+1}=\frac{\sqrt{a_n c_n}+\sqrt{b_n ...
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2answers
167 views

Closed form for the limit of the iterated sequence $a_{n+1}=\frac{\sqrt{(a_n+b_n)(a_n+c_n)}}{2}$

Is there a general closed form or the integral representation for the limit of the sequence: $$a_{n+1}=\frac{\sqrt{(a_n+b_n)(a_n+c_n)}}{2} \\ b_{n+1}=\frac{\sqrt{(b_n+a_n)(b_n+c_n)}}{2} \\ c_{n+1}=\...
7
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1answer
218 views

A new type of Arithmetic-Harmonic mean for $n$ numbers

Let's introduce the following iterative procedure. Take two numbers $x_0$ and $y_0$. $$a_0=\frac{x_0+y_0}{2}~~~~~~~~~~~b_0=\frac{2x_0y_0}{x_0+y_0}$$ $$x_1=\frac{x_0+a_0+b_0}{3}~~~~~~~~~~~y_1=\frac{...
6
votes
4answers
158 views

Show that $\frac{1}{2}\le \sum\limits_{k=0}^n\frac{1}{n+k}\le1$

for $n\in \mathbb N^+$ $$\frac{1}{2n}\le \frac{\frac{1}{n}+\frac{1}{n+1}+...+\frac{1}{n+n}}{n}\le\frac{1}{n}$$ I tried math induction and I tried take integral but I want to solve this with most ...
6
votes
2answers
77 views

May conjecture AM-GM without positivity $a_{1}a_{2}a_{3}\cdots a_{2n} \le\left(\frac{a_{1}+a_{2}+\cdots+a_{2n}}{2n}\right)^{2n}$

Let $a_{1},a_{2},\cdots,a_{2n-1},a_{2n}$be real numbers,I conjecture $$a_{1}a_{2}a_{3}\cdots a_{2n}=\prod_{i=1}^{2n}a_{i}=\le\left(\dfrac{a_{1}+a_{2}+\cdots+a_{2n}}{2n}\right)^{2n}\tag{1}$$ I have ...
6
votes
3answers
224 views

If $\sum_{n=1}^\infty \frac{1}{a_n}$ converges, must $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n}$ converge?

Suppose $\sum_{n=1}^\infty \frac{1}{a_n} = A$ is summable, with $a_n > 0,$ $n = 1,2,3,\cdots.$ How can we prove that $\sum_{n=1}^\infty \frac{n}{a_1 + \dots + a_n}$ is also summable? This question ...
6
votes
1answer
224 views

Prove $\lim\limits_{n \to \infty} \frac{\log (n!)}{n \sqrt[n]{\log 2 \cdot \log 3 \cdots \log n}}=1$

This is close to the ratio of arithmetic to geometric means for logarithms of natural numbers (here $\log$ is the natural logarithm): $$\frac{\log 2+\log 3+\dots + \log n}{n \sqrt[n]{\log 2 \cdot \...
5
votes
3answers
2k views

What is the mathematical meaning of this question?

$a,b,c \in\mathbb{Z}$ and $x\in\mathbb{R}$, then the following expression is always true: $$(x-a)(x-6)+3=(x+b)(x+c)$$ Find the sum of all possible values of $b$. A) $-8$ B) $-12$ ...
5
votes
4answers
4k views

Show that the arithmetic mean is less or equal than the quadratic mean

I tried to solve this for hours but no success. Prove that the arithmetic mean is less or equal than the quadratic mean. I am in front of this form: $$ \left(\frac{a_1 + ... + a_n} { n}\right)^2 \...
5
votes
4answers
604 views

Mean of series is always less than the last element

I'm working on an algorithm's efficiency and wanted to know if there was a way to show the following is true. As an example I did this with $n = 10$ and $n=4$ and it was true. I want to know if it is ...
5
votes
1answer
2k views

Expectation of expectation of indicator function

Is the following correct $$E[E[\mathbb{I}(X)]] = E[\mathbb{I}(X)]$$ I assume that $E[E[X]] = E[X]$, as $E[X]$ is a number and expected value of a constant is a constant, and that $\mathbb{I}(X)$ has ...
5
votes
2answers
290 views

Limit of a sequence :

How do I compute the following limit $$ \lim_{ n\rightarrow \infty }{ { \left (\frac { \sqrt [ n ]{ a } +\sqrt [ n ]{ b } +\sqrt [ n ]{ c } +\sqrt [ n ]{ d } }{ 4 } \right ) }^{ n } } $$ $a,b,c,...