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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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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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Prove that $|X_n-X|^r\to 0 \Rightarrow E(|X_n|^r)\to E(|X|^r) $ (Vitali)

I am reading Vitali theorem here (statement page 4 and proof page 8) I am interested in the following part : Let $0<r<\infty$. Let $X_n$ and $X$ be $L^r(\Omega)$ random variables such that $X_n\...
Laurent Claessens's user avatar
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Mean of a Function: Limit

Let the mean of a function $f(x)$ over the interval $(a,b)$ be defined as: $\bar{f}=\frac{1}{b-a}\int_{a}^{b}f(x)dx$. Is there something such as: $\bar{f}=\displaystyle{\lim_{\epsilon \to 0}}\frac{\...
s28's user avatar
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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: $a+b+c=30$ ...
John Denver Bechayda's user avatar
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MAE and RMSE by groups

Consider five real numbers $A_1$, $A_2$, $A_3$, $B_1$, $B_2$. They are errors, the five differences between real values and estimated values. The MAE(mean absolute error) is $$\text{MAE} = \frac{|A_1|+...
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How many samples to take to get mean value of algorithm runtime within 5% error margin to population mean

I am trying to find a formula to check if the sample size (10) I am using to check the runtime of an algorithm is enough for the current system and system load. I am collecting 10 samples of algorithm ...
Krv Perera's user avatar
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Computing error bars for the geometric mean across multiple datasets

I apologize because my understanding of statistics is somewhat basic. I will try to describe the problem as well as I can. My colleagues and I have a graph where we measure the relative performance of ...
Maxime C's user avatar
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Relation between mean and $\beta$ parameter of discrete Boltzmann distribution with degeneracies

I am trying to compute the mean $\bar{E}$ of a Boltzmann distribution $$ p(\pmb{x}) = \frac{1}{Z} e^{-\beta E(\pmb{x})}, $$ knowing that there is a total of $N$ microstates $\pmb{x}$ and $[\alpha N]$ ...
edfi's user avatar
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I am getting the answer as 4, so where am I making the mistake?

Question: In an examination, the average marks of 4 girls and 6 boys is 24. Each of the girls has the same marks while each of the boys has the same marks. If the marks of any girl is at most double ...
dguy's user avatar
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Minimum squares with two different means?

$X∈U(0,ΞΈ)$ and $π‘Œ ∈ π‘ˆ ( 0 , 4πœƒ )$ . $𝑋$ and $π‘Œ$ are independent. We want to estimate πœƒ using the Least Squares Method and we obtain the outcomes $π‘₯ = 5.2$ and $𝑦 = 28.3$. Determine the Least ...
Need_MathHelp's user avatar
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Power diffs: $\frac{1}{\gamma}\mathbb{E}[X^\gamma - X_*^\gamma] \leq 0 \implies \mathbb{E}[(1+ X^\gamma - X_*^\gamma)^{\frac{1}{\gamma}}] \leq 1$?

I would like to show that the following implication is true for all $\gamma<1$ and $a$ in the $\mathbb{R}^d$ simplex $\Delta^d$: $$\frac{1}{\gamma}\mathbb{E}[(a^TX)^\gamma - ({a^*}^TX)^\gamma] \leq ...
Uomond's user avatar
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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 ...
asdoaihco's user avatar
4 votes
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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 ...
Jiner Soling's user avatar
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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 ...
Starlight's user avatar
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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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Difference between grade averages of different graders

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: ...
Dan's user avatar
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Proving inequality involving mean, covariance and their estimate.

Let $$ A = \left( I - \frac{\Sigma \iota \iota'}{\iota' \Sigma \iota} \right) (\mu - \hat \mu) + \gamma \left( I - \frac{\Sigma \iota \iota'}{\iota' \Sigma \iota} \right)(\hat \Sigma \iota), \...
alejandroll10's user avatar
6 votes
1 answer
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deriving covariance of SDE from fokker-planck

In the book 1 the covariance of an SDE is derived. I am not sure about a particular step. Let me describe it in a TLDR version, then in a longer version. We have an SDE $$dx = f(x,t) dt + L(x,t) d\...
black's user avatar
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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 ...
PVJL's user avatar
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Find $k$ using Chebyshev's inequality such that $E(X)=100,\sigma(X)=10$ and $P(X\le 10k)\ge 4/5$

Given $E(X)=100,\sigma(X)=10$ and $P(X\le 10k)\ge 4/5$, how do we find a lower bound on $k$ using the Chebyshev's inequality Let $X$ be any r.v. with finite mean, $ΞΌ$, and finite variance. Then $βˆ€a &...
Sooraj S's user avatar
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What is the intuition behind the single-pass algorithm (Welford's method) for the corrected sum of squares?

The corrected sum of squares is the sum of squares of the deviations of a set of values about its mean. $$ S = \sum_{i=1}^k\space\space(x_i - \bar x)^2 $$ We can calculate the mean in a streaming ...
Foobar's user avatar
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Let a r.v. $0\leq X \leq 1$. Prove that $V(X) \leq \frac{1}{4}$. When do we have an equality?

Question: Let a r.v. $0\leq X \leq 1$. Prove that $V(X) \leq \frac{1}{4}$. When do we have an equality? Answer: 1- $0\leq X \leq 1 \Rightarrow X^2 \leq X \Rightarrow \int X(\omega)^2 dP (\omega) = E[X^...
OffHakhol's user avatar
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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 ...
joelproko's user avatar
2 votes
2 answers
80 views

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 ...
ACertainArchangel's user avatar
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51 views

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 ...
user2961927's user avatar
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How is it possible for the mean of a log-normal distribution to have units?

I have a set of measurements $a$ (units m) which are log-normally distributed (with parameters $\mu$ and $\sigma$). The expected value (or mean) of $a$ is just the first arithmetic moment, i.e. $$ E(a)...
Plagioclase's user avatar
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1 answer
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Statistical significance with extremely low mean difference with Wilcoxon-Signed Ranked Test

I'm currently conducting an experiment with a relatively large time-based set of data. The data is not normally distributed, so a wilcoxon test is required. I recently compared two parameters for ...
Archonic's user avatar
1 vote
1 answer
48 views

Recursive computation of a weighted mean

Background Let $u=[u_1,\dots,u_N]'$ be a generic vector, its mean value is defined as $$ \bar{u}_N \triangleq \frac{1}{N}\sum_{i=1}^N u_i $$ in few simple steps one can prove the following recursion \...
matteogost's user avatar
1 vote
3 answers
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What is statistically more accurate - the average of a dataset of calculated concentrations or the sum of the total mass divided by the total volume? [closed]

Hopefully an easy question to answer. But if I have a series of five water samples, each sample is a different volume with a different number of plastic particles contained within. I can calculate the ...
E Wisniewski's user avatar
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18 views

General Expression for the t -th difference of conditional means

In econometrics, it is common to work with the difference-in-differences of conditional means. For example, let $Y$ denote a variable of interest and $X_{1}$ and $X_{2}$ denote binary regressors. The ...
Kevin Durant's user avatar
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1 answer
54 views

Does the order of taking the mean across dimensions of a tensor matter?

Given a tensor of data with the shape $(a, b, c)$, I want to aggregate this tensor such that I have a shape of $(a,)$. Now my question is, does it matter whether I first take the mean across dimension ...
hans's user avatar
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Is there a way to express the geometric mean in terms of the $\mathrm{L}^p$ norms?

Sorry, I can't find a good tag for this. So the "scaled" $\mathrm{L}^p$ norm is: $$ \| \mathbf{x} \| = \left( \frac{1}{n} \sum\limits_{j=1}^{n} |x_j|^p \right)^{1/p} $$ and the geometric ...
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Function: average length, partial derivatives

Let $$ d_{a}=\frac{1}{n}\sum_{i=1}^{n-1}d_{i}, $$ be the average length of the polyline given by vertices $(p_{1},...,p_{n})$ where $p_{i}=[x_{i},y_{i}]$ and $d_{i}=\left\Vert p_{i+1}-p_{i}\right\...
justik's user avatar
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Concentration around the median implies concentration around the mean - Theorem 5.1.4 from High-Dimensional Probability [closed]

Let $M$ denote the median of a function $f(X)$ that is Lipschitz continuous with $\left \| f \right \|_{Lip}=1$. I am trying to show that if $\left \| f(X)-M \right \|_{\psi_{2}}\leq C$, then $\left \|...
Moradei's user avatar
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Suppose $X_{n}$ is series of random variables such that it is $\sum_{n=1}^{\infty} E(X_{n}^{4}) < \infty$. Prove that $X_{n} \rightarrow 0$ a.s.

Suppose $X_{n}$ is series of random variables such that it is $\sum_{n=1}^{\infty} E(X_{n}^{4}) < \infty$. According to Markov's inequality we know that $$P(\lvert X_{n} \rvert > \epsilon) \leq \...
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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. ...
Marie Hoffmann's user avatar
1 vote
1 answer
51 views

Comparing proportions inside groups...? [closed]

Let's say I wanted to compare the wealth of people in 2 groups to see in which one the wealth is distributed in a more progressively escalated manner: In group A we have 3 people: One with 12,000 $ ...
vengaq's user avatar
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Can we compare the arithmetic mean of the ratios given the comparison between individual arithmetic means?

I have positive real random numbers $u_1,\ldots,u_n$ and $v_1,\ldots,v_n$ and $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$. I know that the arithmetic mean of $u_i$'s is greater than the arithmetic mean of $...
zdm's user avatar
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Proving $m_4m_2\geq m_3^2+m_2^2$

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] \...
MsBonniePython's user avatar
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 $x_1 = a, x_n = b$ ...
Sajid's user avatar
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1 vote
1 answer
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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 ...
jatrophalouvre's user avatar
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 '...
Nameless's user avatar
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 ...
Heidegger's user avatar
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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 ...
Khosrotash's user avatar
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2 votes
1 answer
50 views

Deducing an incorrect statement when the mean and median are given

$~a,~b,~c,~d$ 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? ...
Darshit Sharma's user avatar
2 votes
1 answer
35 views

Mean speed in a network

I am new to this community so I hope this is the rigth place for this question. I am working on traffic simulations on a certain area, and I need to know which is the average speed in the area during ...
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