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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21 views

How to “denormalize” a standard deviation?

Lets say I have a vector of variables which have all been standard normalized with $(\mu_n, \sigma_n)$ by doing the operation $\frac{\mathbf{y} - \mu_n}{\sigma_n}$. I then have a model which predicts ...
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Is this weighted average smaller than the corresponding arithmetic average?

Is it true that the following weighted-average is smaller than the respective arithmetic average $$ \sum_{n=1}^{N}\left(\frac{b_{n}}{\sum_{m=1}^{N}b_{m}}\right)\frac{a_{n}}{b_{n}}\overset{?}{\le}\frac{...
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Derive the step deviation method for mean

Arithmetic Mean of grouped data can be found by $\bar{x}=\frac{\sum f_ix_i}{\sum f_i}$ where $x_i$ is the midpoint of each class. Aother equation to evaluate Arithmetic Mean of grouped data is using ...
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Can an average of an overall set be equal to the average of a subset?

I read in a publication that the average salary for lawyers in America is $\bar x=\$163,595$. Of these salaries, the average for men is $\bar x_m=\$183,687$, and for women, it is $\bar x_w=\$163,595$. ...
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No sequence of rv's such that $X_n\overset{P}{\rightarrow}0$ and $\mathbb{E}(X_n)\to 2$ and also $\sup\mathbb{E}(X_n^2)<\infty$.

Prove that there is no sequence $(X_n)$ such that $X_n\overset{P}{\rightarrow}0$ and $\mathbb{E}(X_n)\to 2$ and also $\sup_n\mathbb{E}(X_n^2)<\infty.$ Attempt. If we didn't have the last ...
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A machine is used to fill packets with rice. The contents of each packet weighs X grams - Normal Distribution

A machine is used to fill packets with rice. The contents of each packet weighs X grams. X is normally distributed with mean μ grams and standard deviation 3.71 grams. The mean weight μ is stated to ...
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Probability that the mean of a sequence of random variables never exceeds a critical value

Let $p,\mu_c \in [0,1]$. Let $(X_i)_{i \in \mathbb{N}}$ be a sequence of i.i.d random variables, each following a Bernoulli distribution $\mathcal{B}(p)$. For $n \in \mathbb{N}$, let $ \mu_n = \frac{1}...
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If harmonic means of two random variables are equal, does arithmetic means of them are equal too?

If there are two random variables whose harmonic means are equal, does they have same arithmetic mean? In other words, If $E[\frac{1}{X}]=E[\frac{1}{Y}]$, then $E[X]=E[Y]$? Assuming that we know ...
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How do I find the mean from this frequency table with intervals?

This question is from the book "Teach Yourself Statistics" by Richard Goodman: https://archive.org/details/TeachYourselfStatistics The solution provided is: $26.89$ and $45.14$. How do I get this ...
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Analyzing a definition of average which uses a variation of the Lebesgue integral and measure [closed]

Edit: Why was my post downvoted. Please explain how I can improve my post. Second Edit: I changed my Post Introduction Consider $f:A\to[a,b]$ where $A\subseteq[a,b]$ and $S\subseteq A$. I want to ...
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Mean deviation gives better results when deviations are taken from the median, why?

The mean deviation gives better resuls when deviations are taken from the median instead from the mean, because the sum of the deviations from the median is less than the sum of the deviations from ...
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How can we extend this inequality?

Let $a_1, a_2...a_n, b_1, b_2...b_n \in \mathbb{R}^+$. Then, I can prove that, $$ \max_i\frac{a_i}{b_i}\geq\frac{\sum_{i = 1}^n a_i}{\sum_{i = 1}^n b_i} \geq \min_i \frac{a_i}{b_i}$$ Assuming the ...
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Manhattan 5lb, Standard Deviation and Normal Distribution, Question 6 [closed]

the question image Set X consists of 9 total terms, but only two different terms. Six of the terms are each equal to twice the value of each of the remaining 3. Which of the following would provide ...
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Need help to explain the objective function with numerical example.

ai = element in i position in matrix? Cj= column j? Xc(i,j) = X in cluster following (i,j) position in matrix?
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On variations of a claim due to Kaneko in terms of Lehmer means

In this post, for a tuple of positive real numbers $\mathbb{x}=(x_1,x_2,\ldots,x_n)$ we denote its corresponding Lehmer mean as $L_q(\mathbb{x})$, where $q>0$. This is an important example of mean, ...
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centered moments of a random variable being larger than another

Suppose that random variables $X,Y$ have equal means, and that for $k=2,3,\ldots$, the $k$-th centered moment of $X$ is no smaller than that of $Y$. Can we say anything about the relation between $X,Y$...
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Combine two means and two SDs

Hello I would like to combine two means and two standard deviations. The information I have to go on is: Group 1: n=8; mean age 35; SD 9.3 Group 2: n=9; mean age 37; SD 9.6 Any help gratefully ...
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Re-scaling a set of data

I came across this formula and was wondering why we would subtract and divide by the same value, I did my research and it does not seem a normalization technique: Here's the formula
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Lower bound for the arithmetic mean based on quadratic mean

It is known that the arithmetic mean of a list of non-negative real numbers is less than or equal to the quadratic mean (root mean square) of the same list: $$\frac{x_1+x_2+\cdots+x_n}{n} \le \sqrt{\...
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Does $\frac{1}{n}\sum_{k=0}^{n-1}f(k)=f(n)+\epsilon_n$ imply that $f(n)$ converges?

If we let $f(n)$ be an arbitrary arithmetic function with the property that $\forall$ $n>0$ $$\frac{1}{n}\sum_{k=0}^{n-1}f(k)=f(n)+\epsilon_n$$ where $\lim_{n\to\infty}\epsilon_n=0$, then does $\...
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How do I find mean of bus waiting time and what distribution should i use?

I recently am working on an assignment and got stuck in this problem. Say the bus leaves the bus stop with a small amount of random variation due to traffic and washroom breaks etc. The bus departure ...
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Mean of product of two dependent random variables

Let $\boldsymbol{X} \sim Mult_K(n, \boldsymbol{p})$ and $k, l \in \{1, \dots, K\}$. I need $Cov(X^2_k, X^2_l)$, but I am not sure how to calculate $E(X^2_k \cdot X^2_l)$. With the law of iterated ...
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Expectation of Absolute Deviation From Mean

Consider a random variable $X$ and $E[|X|] < 1$. Hence, its expectation $E[X]$ exists. Let us denote $\mu_X := E[X]$ for notational simplicity. The absolute deviation from the mean is $|X-\mu_X|$,...
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How to solve this equation with min function inside sum of multiset?

Let $m$ be a finite multiset of real numbers. Solve for $x\in R$ $$\frac{\sum_{i \in m}{min(x+1,i)}}{|m|} = x$$ In other words, find $x$ such that when you replace all numbers from $m$ that are ...
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Calculate weighted estimated variance [duplicate]

Hello I have to calculate the estimated variance between 3 samples Here is an example ...
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102 views

A weak form of the abc conjecture involving the definition of Hölder mean

I wondered about a weak form of the abc conjecture, see the Wikipedia abc conjecture using the theory of generalized means, I mean this Wikipedia Generalized mean. We get the following claim, where $\...
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1answer
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Can weighted means of functions intersect?

This question came to me as part of a larger problem in game theory I was trying to figure out. $F$ is a family of continuous functions defined in the range $[0,1]$. Initially $F$ contains only the ...
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Problem with an iteration

I have problems to understand the following iterative argument. I don't understand, how the denominators arrise, i.e. how the facultys build. When I plug in the inequality into itself, I get what ...
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How to find the probability of the number of data

I have an exercise, i try to finish all of the question. I stack for the last one. Anyone can help me? I attach the question. I find the solution for (i) 2 (ii) 2.63 (iii) 0.055 And i need your help ...
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Determine point value in quadrilateral

This is a rephrased question after I gained a better understanding of the problem. I have an X,Y plane with one point in each quadrant. Each point represents a value, it is a function result for ...
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How to measure the understanding level of a new concept that was taught to a group of students

Let us say I am an online teacher and I have to introduce a new topic in the class today. Let us say the topic is "Solving a Quadratic Equation". There is an online community of students from ...
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1answer
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Proper way to calculate the mean of a dataset - use the entire dataset or the frequency table values?

I'm not trained in mathematics, but I took a course about using statistics in understanding hotel data. One step of the process requires us to calculate the mean rate of a collection of stay records. ...
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What are these calculation steps in the university entrance exam score calculation in Turkey & what are their purpose?

I'm a teacher (biology). I'm preparing a brochure for students in which I'll explain the calculation method of the score in the university entrance exam (YKS) in Turkey. My main aim is to correct some ...
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Distribution of sample mean squared

I have an iid sample of n Poisson($\theta)$ RVs. I have derived that the MLE for $\phi = \theta^2$ is $\bar{x}^2$. I need to show this estimator is consistent. To show consistency I have to show $Var(...
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Measuring average time

I have a dataset of times (time to execute a task) and I'm now interested in calculating the average time. I might just be mixing stuff up and doing it more complicated than it is but.. The time's ...
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Refinement of an inequality that involves a mean built on the Lambert $W$ function

Deducing a similar derivation which is explained in the Wikipedia Stolarsky mean I define for real numbers $1\leq x<y$ the mean $$M(x,y)=\frac{1}{e}\left(\frac{x-y}{e^{W(x)}-e^{W(y)}}-1\right)\exp\...
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Around a weak form of the Riemann hypothesis inspired in the relationship between the Stolarsky means and the logarithmic mean

Robin's equivalence to the Riemann hypothesis can be written as $$\frac{\sigma(n)-n}{\gamma+\log\log\log n}<M_{\text{lm}}(\sigma(n),n)\tag{1}$$ for enough large $n$ (it is well-know this suitable ...
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Is the harmonic mean directly proportional to the sum of the numbers?

Which of the following sentences is mathematically correct: The harmonic mean of a set of numbers is directly proportional to the sum of numbers. The arithmetic mean of a set of numbers is directly ...
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Effect on variance when 2 observations out of the set of n observations are added and subtracted with a constant

Set $y$ is given such that, $$y= (y_1,y_2,...,y_n)$$ And $$y_1\leq y_2\leq...\leq y_n$$ Now take $y'$ such that $$y'= (y_1,\ y_2,...,\ y_j+c,\ y_{j +1},....,\ y_k-c,\ y_{k +1}...,y_n)$$ ...
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1answer
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Distribution of the sample mean

It is my understanding that when you want to find the distribution of the sample mean of some rv, you can do the following. Let $f(x)$ denote the pdf of $X$: $$f(x)=\frac{e^{-x (\lambda +\mu )} I_1\...
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The arithmetic mean of $2$ numbers creates an arithmetic sequence with them, but how can I apply this for $n$ numbers in general?

If we have two numbers $a$ and $b$ with $a<b$, and let the arithmetic mean of $a$ and $b$ be $A$, we can create an arithmetic sequence {$a, A, b$}. I know that we can't do exactly the same for ...
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Equation relationship between the arithmetic mean, the geometric mean, and the harmonic mean of more than two numbers

Let $A$ be the arithmetic mean of set $s$, $G$ be the geometric mean, and $H$ be the harmonic mean. I know that when there are two terms in a set, $G=\sqrt{AH}$. However, is there an equation like ...
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Best way to determine improvement knowing the mean, standard deviation and standard error of the measures

I have data of a study carried out on two groups of people, one of 13 and the other of 22. Each group was given two measurements, one at the start of the experiment and another one at the end. ...
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1answer
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Knowing the arithmetic, geometric, and harmonic mean of a set, can I find the number of terms in the set? If not, what more information do I need?

Consider a set $a$ with $t$ terms. Knowing the arithmetic mean, the geometric mean, and the harmonic mean of $a$, could I somehow solve for $t$? If not, what more information would I need to know?
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Difference between Bhattacharyya distance and KL divergence

Hello I am new to statistics therefore i would like to ask the following question. I recently encountered the terms Bhattacharyya distance and KL divergence. I also found out that the two measures ...
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1answer
68 views

Conditional mean

Let $X, Y$ be dependent discrete variables. I need to get $E[X|Y]$ and I found a formula $$E[Y∣X=x]=E[Y]+ \text{corr}(X, Y) \frac{\sqrt{\text{var}(Y)}}{\sqrt{\text{var}(X)}}(x −E[X]).$$ I can not ...
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Sums of harmonic means?

This is my first post, so go easy on me if I've made some mistakes. I'm also not a mathematician, but rather an experimental physicist. Basically, I have a problem where I need to take the mean of a ...
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Integral as an average and normalisation

There are some mathematical aspects I do not really understand from my macroeconomics course. For labour $N_t$ and capital $K_t$ following equations are given: $$N_t=\int_{0}^1 N_{jt} dj=1 \qquad (1)...
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Covariance of a random variable and its mean

I want to find the value of $cov(X, \bar{X})$. If I input this in the formula, I will get: $\sum (X - \bar{X})(\bar{X} - \bar{X})/N = 0 \:\:\:\:\:\:(\because \bar{\bar{X}} = \bar{X})$ Edit: Here $X$ ...
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Change of variables for normal distribution

I consider a normal distribution $f(x;\mu_x,\sigma_x)$ with mean $\mu_x = 190$ and standard deviation $\sigma_x =3.08$, which gives me the exemplary value of $f(x=200;\mu_x,\sigma_x) \sim 10^{-5}$. ...

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