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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.

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. Reference: Wikipedia.

For a finite population, the population mean of a property is equal to the arithmetic mean of the given property while considering every member of the population.

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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 proofs on the internet but I was wondering if someone…
Michiel Van Couwenberghe
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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. Now I wonder if there exist distinct real numbers…
Gaurang Tandon
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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 serve as a mean admirably. But then I've noticed…
Yuriy S
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Is it possible to have 2 different but equal size real number sets that have the same mean and standard deviation?

By inspection I notice that Shifting does not change the standard deviation but change mean. {1,3,4} has the same standard deviation as {11,13,14} for example. Sets with the same (or reversed) sequence of adjacent difference have the same standard…
Display Name
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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 arithmetic mean of its elements, and collect the…
Vladimir Reshetnikov
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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.
user1001001
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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 = \frac{1}{\frac{1}{x}-\frac{1}{y}} =…
MathTrain
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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, when the set is organized in ascending or descending…
Mandelbrot
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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. What is the expected number of throws (including…
WaveX
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Why is the geometric mean less sensitive to outliers than the arithmetic mean?

It’s well known that the geometric mean of a set of positive numbers is less sensitive to outliers than the arithmetic mean. It’s easy to see this by example, but is there a deeper theoretical reason for this? How would I go about “proving” that…
TheProofIsTrivium
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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 involves the difference of the arithmetic and…
zahbaz
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For what values does the geothmetic meandian converge?

The geothmetic meandian, $G_{MDN}$ is defined in this XKCD as $$F(x_1, x_2, ..., x_n) = \left(\frac{x_1 +x_2+\cdots+x_n}{n}, \sqrt[n]{x_1 x_2 \cdots x_n}, x_{\frac{n+1}{2}} \right)$$ $$G_{MDN}(x_1, x_2, \ldots, x_n) = F(F(F(\ldots F(x_1, x_2,…
Pro Q
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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 of elements, gives you the sum of all the elements.…
DLV
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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 $Q$: From this picture the ordering of these power…
Semiclassical
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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 Mathematica, but I can't figure out how to prove it. The…
Yuriy S
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