# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### 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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### 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,...
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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, ...
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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 ...
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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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### 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 ...