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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Where have people discussed "higher" arithmetic-geometric means?
The arithmetic-geometric mean of two nonnegative real numbers $x,y$ is obtained by taking their arithmetic and geometric mean
$$ a_1 = \frac{1}{2}(x + y), \quad g_1 = (xy)^{\frac{1}{2}} $$
and then ...
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Generalising power mean to negative numbers
I've been researching generalisations of the arithmetic mean and have come across the definition of a power mean (or generalised mean),
$$
M_p(x) = \left(\frac{1}{n}\sum_{i=1}^{n}x_i^p\right)^{1/p}.
$$...
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Standard deviation tells how a data is spread around mean. Then what is 1 SD (68%), 2SD (95%)?
I am learning statistics from scratch. I understand that SD would tell how datapoints are spread around mean. However, I do not understand how it is useful in finding outliers. Especially, these two ...
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Calculating the Uncertainty in a Rolling-Averaged Time Series
I have a time series of data for the heat absorbed by the ocean (in Joules) since the 1957, downloaded from the NASA website:
The data consists of a series of measurements as well the standard error ...
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"Partial limits" and rates of convergence
As per Wikipedia,
A sequence $(x_{n})$ that converges to $L$ is said to have order of convergence $ q\geq 1$ and rate of convergence $\mu\in (0,1)$ if
\begin{equation}\lim _{n\rightarrow \infty }{\...
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Inequalities for weighted $\ell^p$ norms?
Consider the $p$-norm with weight factors over $\mathbb R^n$:
$$
\| x \|_{p,\lambda} := \Big( \lambda_1 |x_1|^p + \lambda_2 |x_2|^p + \dots + \lambda_n |x_n|^p \Big)^{\frac 1 p}
$$
Here, $\lambda_i &...
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Find maximum likelihood estimators of the mean and variance a given distribution
I am given a density function $f_X(x|a)$.
Given a random sample $x_1, x_2, \cdots , x_n$, I calculated the maximum likelihood estimator for the variable $a$. Let's call it $\bar{a}$.
Now, I was asked ...
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A Taylor Series for the Arithmetic-Geometric Mean
I let $f(x)=\operatorname{AGM}(1,1+x)$ on Wolfram Alpha and used various small values of $x$ to estimate $f'(0)$, $f''(0)$, and $f'''(0)$. I came up with $1/2$, $1/8$ and $5/27$, leading to $f(x)$ ...
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Is the geometric mean of two numbers unique?
From my textbook:
Prove that if $a, b$ and $c$ are three successive terms of a geometric sequence, then $b^2=ac$ $(b$ is called geometric mean of $a$ and $c).$
Is this exercise incorrect? I think ...
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Bounds for means generalized to ratio of integrals
Despite not being sure about the origin, I found the following inequality for means in here, that is
$$\min_i \frac{|x_i|}{|y_i|}\leq \frac{\sum_i |x_i|}{\sum_i |y_i|} \leq \max_i \frac{|x_i|}{|y_i|} $...
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Unbiased estimator for sample mean or variance using non-uniform sampling
Wikipedia mentions that given as input n iid samples $Y_1,\ldots,Y_n$, the unbiased sample variance is given as $\frac{1}{(n-1)} \sum_{i=1}^n (Y_i-\mu)^2$, where $\mu$ denotes the sample mean.
I was ...
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What is the probability that a randomly chosen number from the $10$ numbers would be more than $7$?
The average of $10$ numbers is $7.5$.
What is the probability that a randomly chosen number from the $10$ numbers would be more than $7$?
This is the response from the author of the question.
$$\begin{...
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Is $x>y$ or $x<y$ or $x=y$, or it differs in different situations?
I can't understand the difference between these 2 questions that lead to different answers!
Q1. List $\text{A}$ consists of $x$ numbers, and the sum of all numbers of $\text{A}$ is $52$. List $\text{B}...
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Prove that $A \leq G^2$, where $A$ is the arithmetic mean and $G$ is the geometric mean.
I know that $G \leq A$ where $G = \sqrt[n]{a_1 \cdot a_2 \cdot \cdot \cdot a_n} \ $ and $ \ A = \frac{a_1+a_2+...+a_n}{n} \ $ and $ \ 1 \leq a_1, a_2, ... , a_n \leq n$ (this is given to me for this ...
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If $x+y=3\sqrt{xy}$, $\frac xy=\frac{a+\sqrt b}{a-\sqrt b}$, given $x,y>0$ and $a,b$ are coprime, then $a+b=?$
If the sum of two numbers $x,y$ is $3$ times their geometric mean and $\frac xy=\frac{a+\sqrt b}{a-\sqrt b}$, given $x,y>0$ and $a,b$ are coprime, then find $a+b$
My Attempt:
$x+y=3\sqrt{xy}\...
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How can I determine the total mixed variable required to reach to a specific mean?
To maintain a population (neither increase nor decrease) the average child per woman needs to be $2.1$.
I want to work out, for every woman that has $2$ children, how many need to have $3$ children, ...
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For which sets of functions can we establish $f$-mean inequality?
For a finite set $X \in \mathbb{R}$ and a bijective increasing function $f:\mathbb{R} \to \mathbb{R}$, let's define the $f$-mean of $X$ as the following.
$$\mu_f(X) \equiv f^{-1} \left( \frac{1}{|X|} \...
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Beyond the $agm(1,x)$ ; $f(a,b) = f((a+b)/2,((a^{1/3} + b^{1/3})/2)^3) $
The Arithmetic-GeometricMean (agm) can be expressed by the CompleteEllipticIntegraloftheFirstKind.
$$agm(a,b) = agm((a+b)/2,\sqrt {ab})$$
$$agm(a,b) = \frac{(a+b)\pi}{4 K(\frac{a-b}{a+b})}$$
where K ...
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Show that $ \frac{S_n}{D_n} \xrightarrow{d} Y \sim N(0,1) \text{as } n \to \infty $ with CLT
Let $(X_n)$, $n \in \mathbb{N}$, be a sequence of iid variables with $\text{E}[X_1] = 0$ and $\text{Var}(X_1) = 1$.
Define the triangular array $(Z_{nk})$, ${1 \leq k \leq n, n \in \mathbb{N}}$, by $...
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What's the intuition behind the mean square of something that deviates around an average being greater than the square of the mean? [closed]
I think I can understand this algebraically but is there a way to see this more intuitively, like understanding why the sum is bigger when the inside is squared compared to when the whole thing is ...
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Maximisation of product
a , b are real numbers such that $2a^2 + 5b^2 = 20$. Then the maximum value of $a^4b^6$ is..
I tried to solve it through the Arithmetic mean $\geq$ Geometric mean principle.
So we suppose
$$
\begin{...
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Expected value of a time series
I'm reading the fourth edition of Shumway and Stoffer's "Time Series Analysis and Its Applications" and I got stuck trying to determine the expected value or mean function of a simple time ...
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Relationship between median and mean of a left skewed distribution
Question
If I see graphically that my distribution is left-skewed. How to concretely (still qualitatively) conclude that the Mean is less than the Median? At least in nonpathological examples?
There ...
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Why is standard deviation important when comparing scores across different distributions?
Initial Question:
Let's say you are a manager, and you have to compare the evaluation of two employees, A and B, who work in different departments—Marketing and Sales, respectively.
...
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How to Find the Maximum Possible Value of the 90th Percentile Given a Fixed Mean?
Problem:
If we have a list where the minimum element is -10 and the mean of the list is 55.2, what is the maximum possible value the 90th percentile could be?
My Approach So Far:
I understand that the ...
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Proof of a definite integral whose result is the difference of arithmetic and geometric means
This might have been already asked in this site but I can't find it. So here's the integral:
$$\int_{r_\text{min}}^{r_\text{max}} \sqrt{\left(1-\frac{r_\text{min}}{r}\right)\left(\frac{r_\text{max}}{r}...
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Correct way to calculate the mean of SO(3) rotation matrices
I have N SE(3) matrices that define the rigid body transformation between a LiDAR and an IMU. These SE(3) matrices are estimated using a calibration procedure at different trials. I want to find the ...
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Standard deviation of the norm of the random vector
Let's say I have $n$ iid random variables $v_i \sim \mathcal{N}(0,c)$ with $c$ is a constant, and put them into a vector: $V=\begin{bmatrix}
v_1\\
v_2\\
...\\
v_n
\end{bmatrix}$. I am interested in ...
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Prove that $|X_n-X|^r\to 0 \Rightarrow E(|X_n|^r)\to E(|X|^r) $ (Vitali)
I am reading Vitali theorem here
(statement page 4 and proof page 8)
I am interested in the following part :
Let $0<r<\infty$. Let $X_n$ and $X$ be $L^r(\Omega)$ random variables such that
$X_n\...
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Mean of a Function: Limit
Let the mean of a function $f(x)$ over the interval $(a,b)$ be defined as:
$\bar{f}=\frac{1}{b-a}\int_{a}^{b}f(x)dx$.
Is there something such as:
$\bar{f}=\displaystyle{\lim_{\epsilon \to 0}}\frac{\...
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Getting the Harmonic Mean of Three Numbers from Arithmetic and Geometric Mean
I have been stuck to this problem:
Given three numbers a, b, c, their arithmetic mean which is 10, and their geometric mean which is 8, find the value of the harmonic mean.
I now know that:
$a+b+c=30$
...
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MAE and RMSE by groups
Consider five real numbers $A_1$, $A_2$, $A_3$, $B_1$, $B_2$. They are errors, the five differences between real values and estimated values. The MAE(mean absolute error) is
$$\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 ...
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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 ...
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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 $[\alpha N]$ ...
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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 ...
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Minimum squares with two different means?
$X∈U(0,θ)$ and $𝑌 ∈ 𝑈 ( 0 , 4𝜃 )$ . $𝑋$ and $𝑌$ are independent. We want to estimate 𝜃 using the Least Squares Method and we obtain the outcomes $𝑥 = 5.2$ and $𝑦 = 28.3$. Determine the Least ...
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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 $\Delta^d$:
$$\frac{1}{\gamma}\mathbb{E}[(a^TX)^\gamma - ({a^*}^TX)^\gamma] \leq ...
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Wikipedia: Convex combination vs weighted average
Wikipedia: Convex combination
a convex combination is a linear combination of points (which can be vectors, scalars, or more generally points in an affine space) where all coefficients are non-...
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balls in a bag probability question
preface: this is a H.W question and I'd be happy to get guidance and get to the answer myself instead of getting the answer from you, I'm struggling with this kind of question partially because I'm ...
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Expected value of a transformed variable.
I'm new to probability and I want to know the general formula for calculating the mean and variance of a transformed random variable.
Let $X$ be a continuous random variable with distribution function ...
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Mean of a random variable in stochastic process.
I'm reading the book "Random Data: Analysis and Measurement Procedures" of "Julius S. Bendat" and "Allan G. Piersol". In this book, a stochastic process is denoted by a ...
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What distribution does the height of both men and women follow?
It is often said that the height of men and that of women follow normal distribution with different means and variances. As graphs suggest, it appears true.
Then, what is the whole distribution of ...
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Change of Limit in Calculation of Mean
This video here has
$CDF=F(x)=\text{sin } x, 0<x<\frac{\pi}{2}$
which gives on differentiating:
$$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}}.
$$
In the Lower bound ...
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Difference between grade averages of different graders
I am a teacher and we now had our end-of-semester exam in which $N$ students participated.
I have $k$ graders each graded a certain amount of exams $N_i$, so $\sum_{i=1}^kN_i=N$.
When I check the ...
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Is there a closed-form expression for this iterated mean?
Here is a simple Python implementation of the arithmetic, geometric, and harmonic means of a (non-empty) list of numbers:
...
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Proving inequality involving mean, covariance and their estimate.
Let $$ 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), \...
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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\...
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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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