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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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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 $$ \sqrt{xy}≤ \frac{x+y}{2} .$$ ...
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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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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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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 ...
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2answers
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Question about arithmetic–geometric mean [duplicate]

We have two sequences: $$a_{n+1}=\sqrt{a_nb_n}$$ $$b_{n+1}=\frac{a_n+b_n}{2}$$ I need to prove that those are making Cantor's Lemma.(At the end I shold get that: $\lim_{n\to \infty}a_n=\lim_{n\to \...
11
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1answer
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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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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 \...
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2answers
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Using strong induction to get the AM-GM inequality for $2^n$ numbers

The arithmetic mean of $k$ numbers $a_1, a_2, \ldots, a_k$ is their average $\frac{a_1+a_2+\cdots+a_k}{k}=AM$. Their geometric mean is $\sqrt[k]{a_1a_2\cdots a_k}=GM$. I am asked to show this: Use ...
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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 = ...
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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 ...
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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 ...
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1answer
826 views

Proving AM-GM with induction

I am trying to prove AM-GM with the following steps: Prove that AM-GM holds for two variables Prove that if AM-GM holds for $k$ variables then it holds for $2k$ variables Prove that if AM-GM holds ...
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3answers
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Proof the the Arithmetic-Harmonic Mean is expressible as the Geometric Mean

We define the Arithmetic-Harmonic mean of $a,b \in \mathbb{R_+}$ such that \begin{gather*} a_{n+1} = \frac{1}{2}(a_n + b_n) \\ b_{n+1} = \frac{2a_{n}b_{n}}{a_{n} + b_{n}} \end{gather*} Let us also ...
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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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1answer
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Prove an algorithm for logarithmic mean $\lim_{n \to \infty} a_n=\lim_{n \to \infty} b_n=\frac{a_0-b_0}{\ln a_0-\ln b_0}$

Take: $$a_0=x,~~~~b_0=y$$ $$a_{n+1}=\frac{a_n+\sqrt{a_nb_n}}{2},~~~~b_{n+1}=\frac{b_n+\sqrt{a_nb_n}}{2}$$ Then we obtain as a limit the logarithmic mean of $x,y$: $$\lim_{n \to \infty} a_n=\lim_{n ...
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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 ...
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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. ...
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0answers
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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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2answers
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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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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 ...
5
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4answers
885 views

How can I Prove $\frac{2xy}{x+y}\leq \sqrt{xy}\leq \frac{x+y}{2}$

for $x,y>0$ prove that $\frac{2xy}{x+y}\leq \sqrt{xy}\leq \frac{x+y}{2}$ I have tried to develop $(x+y)^2=$ and to get to an expression that must be bigger than those above Thanks!
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4answers
242 views

Geometric proof of $QM \ge AM$

Prove by geometric reasoning that: $$\sqrt{\frac{a^2 + b^2}{2}} \ge \frac{a + b}{2}$$ The proof should be different than one well known from Wikipedia: DISCLAIMER: I think I devised such proof (...
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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 ...
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2answers
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QM-AM-GM-HM proof help

Out of interest, I am trying to proof QM-AM-GM-HM inequality. If you don't know it, it's something like this... Let there be $n$ numbers $x_1, x_2, x_3...x_n$, where $x_1, x_2, ...,x_n>0$. Proof ...
3
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1answer
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Convergence in probability implies Fatou's lemma?

Let $(\Omega, \mathcal{F},P)$ be a probability space and $(X_n)$ be a positive-valued sequence of random variables on $\Omega$. We assume that $(X_n)$ converges in probability to the random variable $...
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7answers
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Elementary proof for $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$

I am searching for an elementary proof of the AM-GM inequality in three variables: $\sqrt[3]{xyz} \leq \dfrac{x+y+z}{3}$ The inequality of the geometric mean vs the arithmetic mean of two variables ...
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1answer
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Expected value of lognormal distribution.

Hi I'm stuck on this question: Recall that $X$ is said to have a lognormal distribution with parameters $\mu$ and $\sigma^2$ if $\log(X)$ is normal with mean $\mu$ and variance $\sigma^2$. Suppose $...
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1answer
132 views

Let $a>0$ and $b>0$. Prove that $\sqrt{ab} \le (a+b)/2$. [duplicate]

Let $a>0$ and $b>0$. Prove that $\sqrt{ab} \le (a+b)/2$. Here is what I have tried: Let $a \le b$. Multiplying both sides of this inequality by $a$ results in $a^2 \le ab$. It follows that $a \...
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0answers
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Arithmetic-quadratic mean and other “means by limits of means”

For $x,y$ positive real numbers, and $p\neq 0$ real, define the Hölder $p$-mean $$M_p(x,y) := \left(\frac{x^p+y^p}{2}\right)^{1/p}$$ whereas $$M_0(x,y) := \sqrt{xy}$$ is the limit of $M_p(x,y)$ when $...
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1answer
227 views

how to calculate what I need for final exam

Evaluation: A final mark out of 100 will be calculated as follows: Clicker Questions: 5% WeBWorK Assignments: 10% Midterm Tests (25% of the higher mark + 10% of the lower): 35% Test 1 – Saturday, ...
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1answer
375 views

Expected value and sum of independent variables.

EDIT: I've found my mistake. Flipped around the values because in my head I had them tails up at the start.. Not sure what to do with the question now... On a table there are three coins in a row, ...
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2answers
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Two different formulas for standard error of difference between two means

I mostly see this formula when searching for a formula for the estimate of the standard error in difference between two means, and it is also used in this video. $$\Delta=\sqrt{\left(\dfrac{s_1}{N_1}\...
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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. ...
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3answers
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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 ...
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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 ...
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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,...
7
votes
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{...
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votes
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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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_{...
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 ...
5
votes
1answer
581 views

Question about Geometric-Harmonic Mean.

Define our Harmonic sequence for two numbers such that \begin{equation} a_{n+1} = \frac{2a_nb_n}{a_n + b_n} \end{equation} and our geometric sequence \begin{equation}b_{n+1} = \sqrt{a_nb_n} \end{...
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,...
5
votes
2answers
743 views

Inequalities involving arithmetic, geometric and harmonic means

Let $A$, $G$ and $H$ denote the arithmetic, geometric and harmonic means of a set of $n$ values. It is well-known that $A$, $G$, and $H$ satisfy $$ A \ge G \ge H$$ regardless of the value $n$. ...
4
votes
3answers
243 views

How to go upon proving $\frac{x+y}2 \ge \sqrt{xy}$? [duplicate]

I'm trying to prove this but am having some difficulty. For any $x,y\in\mathbb R$ such that $x\ge 0$ and $y\ge 0$ we have $$\frac{x+y}2 \ge \sqrt{xy}.$$ So far what I have gotten to is $\frac{x+y}{...
4
votes
2answers
722 views

Intuition behind the Geometric Mean

Our (awesome) statistics professor told us about the best intuition behind the definition of the Arithmetic Mean (he had heard of during his career). Here's what he said: Imagine a 10-yard-long ...
1
vote
2answers
53 views

Calculation of probability with arithmetic mean of random variables

There are 4 people, each of whom has one deck of cards with 500 cards that are numbered from 1 to 500 with no duplicates. Each person draws a card from his deck and I would like to calculate the ...
7
votes
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}=\...
4
votes
2answers
226 views

Extending the ordered sequence of 'three-number means' beyond AM, GM and HM

I want to create an ordered sequence of various 'three-number means' with as many different elements in it as possible. So far I've got $12$ ($8$ unusual ones are highlighted): $$\sqrt{\frac{x^2+y^2+...
4
votes
1answer
402 views

Computing square roots with arithmetic-harmonic mean

We know that if we iterate arithmetic and harmonic means of two numbers, we get their geometric mean. So, basically if we need to compute the square root of $x$: $$\sqrt{x}=\sqrt{1 \cdot x}=AHM(1,x)$...