# 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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### Standard deviation of the norm of the random vector

Let's say I have $n$ iid random variables $v_i \sim \mathcal{N}(0,c)$ with $c$ is a constant, and put them into a vector: $V=\begin{bmatrix} v_1\\ v_2\\ ...\\ v_n \end{bmatrix}$. I am interested in ...
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### Getting the Harmonic Mean of Three Numbers from Arithmetic and Geometric Mean

I have been stuck to this problem: Given three numbers a, b, c, their arithmetic mean which is 10, and their geometric mean which is 8, find the value of the harmonic mean. I now know that: $a+b+c=30$ ...
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### Wikipedia: Convex combination vs weighted average

Wikipedia: Convex combination a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-...
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### balls in a bag probability question

preface: this is a H.W question and I'd be happy to get guidance and get to the answer myself instead of getting the answer from you, I'm struggling with this kind of question partially because I'm ...
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### Expected value of a transformed variable.

I'm new to probability and I want to know the general formula for calculating the mean and variance of a transformed random variable. Let $X$ be a continuous random variable with distribution function ...
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### Mean of a random variable in stochastic process.

I'm reading the book "Random Data: Analysis and Measurement Procedures" of "Julius S. Bendat" and "Allan G. Piersol". In this book, a stochastic process is denoted by a ...
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### What distribution does the height of both men and women follow?

It is often said that the height of men and that of women follow normal distribution with different means and variances. As graphs suggest, it appears true. Then, what is the whole distribution of ...
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### Change of Limit in Calculation of Mean

This video here has $CDF=F(x)=\text{sin } x, 0<x<\frac{\pi}{2}$ which gives on differentiating: $$PDF=f(x)=\text{cos } x, 0<x<\frac{\pi}{2}$$ But then when calculating the mean is there a ...
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### Under what conditions is the arithmetic mean an upper bound for the quadratic mean?

As is well known, the arithmetic mean (AM) is less than or equal to the quadratic mean(QM), i.e., $$\frac{x_1+x_2+\cdots+x_n}{n} \leq \sqrt{\frac{x_1^2+x_2^2+\cdots+x_n^2}{n}}.$$ In the Lower bound ...
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I am a teacher and we now had our end-of-semester exam in which $N$ students participated. I have $k$ graders each graded a certain amount of exams $N_i$, so $\sum_{i=1}^kN_i=N$. When I check the ...
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### Is there a closed-form expression for this iterated mean?

Here is a simple Python implementation of the arithmetic, geometric, and harmonic means of a (non-empty) list of numbers: ...
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### RMS/quadratic mean and confidence interval

After searching on the web without success, I'm asking for help here. I wasn't taught statistics, and I get lost in a lot of formulas that I often find hard to understand. I'm also not used to posting ...
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### In what situations would using means other than arithmetic/geometric/harmonic make sense?

I understand some use cases for arithmetic (standard), geometric (average growth of two successive discrete growth rates), harmonic (average velocity when consecutively traveling the same distance ...
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### Rigorous proof that $1/\sum_ir_i^{-1}\le\min r_i$

I am taking an introductory physics course where the equivalent resistance of resistors in parallel is: $$\bigg ( \sum_{i=0}^n \frac 1{r_i} \bigg ) ^{-1} = R_{eq}$$ My book says that $R_{eq}$ will ...
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### Prove or give counter example of quadratic inequality

I have two finite probability mass functions (pmfs) $P(x)$ and $Q(x)$ on the same support $(0,1,\ldots,n)$. Let $(p_0,p_1,\ldots,p_n)$ and $(q_0,q_1,\ldots,q_n)$ be the probability vectors from the ...
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### Aggregate for Entropy over Cohort

Assume I have a black-box predictor that behaves stochastically and I want to capture its uncertainty wrt categorical output $c_j \in \{c_1, c_2, ..., c_J\}$ when given the same input multiple times. ...
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I've been trying to prove: $m_4m_2\geq m_3^2+m_2^2$ How I proceeded: From the Cauchy-Schwarz inequality: $m_3=E[(x-\bar{x})^3]$ $m_2=E[(x-\bar{x})^2]$ $m_4=E[(x-\bar{x})^4]$ $[E(XY)]^2 \leq E[X^2] \... 0 votes 2 answers 57 views ### Arithmetic Mean for Continuous Functions Arithmetic Mean is defined by: $$\bar{x} = \frac{\sum_{i=1}^{n} f(x_i)}{n} = \sum_{i=1}^{n} \frac{f(x_i)}{n}$$ Now, let's define the$x_i$as,$x_{i+1}-x_{i} = \Delta x$with$x_1 = a, x_n = b$... • 85 1 vote 1 answer 100 views ### Which integer sets are closed under integer means$\frac{x+y}{2}$? Question: A set$A\subseteq\{1, 2, \dots, n\}$satisfies: if$x, y\in A$and$\frac{x+y}{2}\in\mathbb{Z}$, then$\frac{x+y}{2}\in A$. Find the number of such$A$in terms of$n$. I thought of this ... • 3,078 2 votes 1 answer 97 views ### Law of unconscious statistician without measure theory I am studying the theory of probability with 'Jaynes style', i.e. not using all the theoretical substratum of measure theory and Lebesgue integral. I would like to understand if there exists in this '... • 79 1 vote 2 answers 59 views ### Creating a new mean I was wondering: are there some necessary criteria to be respected and fulfilled for creating a new statistical mean? This question came up to my mind while studying arithmeticl mean, gometric mean ... • 3,482 0 votes 0 answers 48 views ### What are the mean((s) for numbers between$(0,1)$I am looking for Arithmetic mean - Harmonic mean - geometric mean and root mean square for the numbers in$(0,1)$. Am I doing it right? As the first step, I take a partition for the numbers$\{\frac ...
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$~a,~b,~c,~d$ are positive integers such that $a~<~b~<~c~<~d$ If mean and median of $~a, ~b, ~c,~d$ are $35$ and $39$ respectively, then which one of the following statements cannot be true? ...