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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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Best Average to Represent Quantitative Data?

For a final assignment, my two group members and I have created a biased survey called Should Public Schools Implement a Uniform Policy? We have about 2-3 ...
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Do two pairs of distinct natural numbers exist such that AGM(A,B) equal to AGM(C,D)?

Here AGM is arithmetic-geometric mean. Are there natural numbers A,B,C,D such that $1\leq A<C<D<B$ and arithmetic-geometric mean AGM(A,B)=AGM(C,D) ? In other words, is AGM a homomorphism ...
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Can arithmetic-geometric mean of two distinct natural numbers be a natural number?

Are there solutions to equation $AGM(A,B)=C$ such that $1\leq A<B$ and $A,B,C\in\mathbb{N}$?
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Mean vector and covariance matrix

I am given a home work for one subject, but my probability theory course is just started, so I dont have enough information. Could someone help me with that? Given: $$\begin{equation} p_\underline x(...
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Sampling uncertainty (uncertainty in standard deviation)

Say I took twenty measurements of the wind velocity at a point. This is a sample of the infinite number of measurements I would need to get the true velocity at that point. I then calculate the mean ...
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What is the mathematical meaning of this question?

$a,b,c \in\mathbb{Z}$ and $x\in\mathbb{R}$, then the following expression is always true: $$(x-a)(x-6)+3=(x+b)(x+c)$$ Find the sum of all possible values of $b$. A) $-8$ B) $-12$ ...
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Range of Data Values That Includes 95% of my Data

For a math assignment, I am required to find the range of data values that would include 95% of my data. My data is the wait time for a specific bus. I have gathered 20 pieces of data for this, ...
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Generalized AM-GM Inequality

I was discussing means with my friend, and I tried to illustrate the concept of geometric mean using the following idea: Suppose we have two positive quantities $x,y>0$. The simplest geometric ...
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Statistical problem: Scores, Weights and Averages

As part of a software development project, I'm trying to find the correct mathematical approach for a statistics problem involving scores, weights and averages. Here's a simplified summary of the ...
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Hypothesis Testing for Probability and Statistics

Question: The present defective rate of a product in a certain factory is 3%. A random sample of 250 of products produced using a new technique which is supposed to improve the quality of production ...
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Generalized definition of all means of $n$ numbers?

Following chapter 2 of Greek Means and the arithmetic-geometric mean by George and Silvia Toader, the Greek (or Pythagorean) means of two numbers have been studied for centuries originating in the ...
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What questions lead to different representative statistics?

What are the questions that are asked whose answer yields the geometric and harmonic means. For example, we arrive at the arithmetic mean by solving: $$ \min_y \sum_{x \in S} (x - y)^2$$ What ...
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Which mean to use to find the average of measurements given in the form of “per period”

Let's say I made several measurements on my gas consumption. $2$ m$^3$ per day $1$ m$^3$ per day $1.5$ m$^3$ per day $2$ m$^3$ per day What is the average daily consumption of gas? I can take ...
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The average of the items of $(0,10]$

The question says it. Let $A = \{\,x \,|\, 0 < x \le 10\,\}$. What would be the average of all the items in this set? How do you prove it? UPDATE $x$ belongs to the set of real numbers.My thoughts:...
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Prove / Disprove that $E\big((X−a)^4\big)$ is minimized when $a=E(X)$.

Let $X$ be a random variable with moment up to order $4$. Prove / disprove that $E\big((X−a)^4\big)$ is minimized when $a=E(X)$. I've already proved that $E((X−a)^2)$ is minimized as such, but I'm ...
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What can I say about expected value of expected value of x given y: $E[E[X|Y]]$

My question is about Y that is discrete and for some random variable X, but if its having a meaning for Y that is continuous then please give that case your attention. What can I say about $E[E[X|Y]]...
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Definition of conditional mean of $ E[g(Y)X|Y=y]$ where Y is discrete.

I got the following question: $X$ and $Y$ are random variables upon the probability space $(S,F,P)$. $Y$ is discrete with range $W$. Let $g: \Bbb R\to \Bbb R$ be a function so that $g(Y)$ is a ...
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What is the probability of getting not more than 500 utopians?

The Utopia population: 602427 inhabitants of which 313839 are of utopian nationality. We choose 1000 inhabitants. What is the probability of getting 1) not more than 500 Utopians 2) at least 400 ...
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Does $\alpha_j-(\Pi \alpha_i)^{1/d}=\beta_j-(\Pi \beta_i)^{1/d} $ force $\alpha_i=\beta_i$?

Let $d>1$, and let $\alpha_1,\dots,\alpha_d,\beta_1,\dots,\beta_d$ be positive real numbers. Is there an easy proof of the following claim: If $\alpha_j-(\Pi_{i=1}^d \alpha_i)^{1/d}=\beta_j-(\...
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How to find the probability of a 'mean shift' in a random process?

How would I go about calculating the probability that the observed mean of a collection of data generated by a random process, which is governed by a particular statistical model, would deviate from ...
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Why the standard deviation of discrete random variable is not divided by n?

The standard deviation formula actually makes sense to me. $\sqrt(\frac1N\sum(x-u)^2)$ However, I do not understand why in the standard deviation of a discrete random variable formula $\sqrt( \sum(...
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Compute mean and covariance matrix of $\bar{X}$ from a simple random sample

Given $\{X_\alpha , \alpha =1,...N\}$ a simple random sample obtained from any p-dimensional distribution with mean $\mu$ and covariance matrix $\Sigma$, compute the mean and the covariance matrix of $...
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Does $ \arg \min_{x_{2n}} \sum_{i = 1}^{N} \left( {s}_{i} - x_{2n} \right)^{2n} $, for $n>1$, have a name and/or application?

Given a list of $N$ real numbers $s_i$: the median $x_0$ is $$ \arg \min_{x_0} \sum_{i = 1}^{N} \left| {s}_{i} - x_0 \right| $$ the mean $x_2$ is $$ \arg \min_{x_2} \sum_{i = 1}^{N} \left( {s}_{i}...
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find sample for collection of samples

i have following problem, To estimate the average alcohol consumption of UCSD students, we take three random samples of 40, 45 and 50 students respectively, and their sample means turn out to be 3.15,...
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How can I find the mean value of an exponential profile?

assume a surface density profile \Sigma(y)=\Sigma_p e^{-2y}. How can I find a (fixed) mean value to approximate the mentioned profile? Is it true fo find the value ...
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The mean of four numbers is $12$. When the greatest number is removed, the mean becomes $9$. What number was removed? [closed]

I'm Akash, a nine-year-old fourth-grader living in San Angelo, Texas. Here's a question that I couldn't solve, apparently: The mean of four numbers is $12$. When the greatest number is removed, the ...
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PDF of combination of i.i.d. standard normal variables.

I assume that $a_i$,$b_i$, and $c_i$ are independent standard normal variables by each other ($a_i$, $b_i$, $c_i$ ~ $N(0,1)$). and equation which I want to get PDF or mean and variance is $$\frac{\...
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How to group a sample according to the mean and standard deviation

My research is not about statistics but my tutor told me I need to group my evaluated results in blocks using the standard deviation and the mean. I asked him the details but he's taking a lot of time ...
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mean vs median geometric interpretation?

I'm looking at this picture on wikipedia, comparing the median and mean of an arbitrary distribution. But what does it mean exactly? From the figure, it looks like the mean is the center of mass of ...
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Expected value and variance of step of random walk with barriers

I have to simulate the following game Suppose that two players A and B each start with a stake of \$5, and bet \$0.5 on consecutive coin flips. The game ends when either one of the players has won ...
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Derivative in mean

I have one question about calculation rules for means. Say I have an equation of the form \begin{align*} \langle H(a,b) \rangle = \langle H_1(a,b) \rangle + \langle H_2(a,b)\rangle \end{align*} ...
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Given some statistical measures, reconstruct a list of numbers

Suppose I have a list of numbers $(y_1, y_2, y_3, \dots, y_N)$ with these properties: $$ \sum_{i=1}^{N}y_i = 13\, 776\, 663, $$ $$ \bar{y} = \dfrac{1}{N} \sum_{i=1}^{N}y_i = 17\,135, $$ $$ s^2 = \...
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Calculation of probability with arithmetic mean of random variables

There are 4 people, each of whom has one deck of cards with 500 cards that are numbered from 1 to 500 with no duplicates. Each person draws a card from his deck and I would like to calculate the ...
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Why is the expectation of cauchy distribution not defined? (What is the intuition behind it?)

Let $X$ be random variable with pdf $f_X(x) = \dfrac{1}{\pi(1+x^2)}$. I understand that mathematically, the improper integral, $\displaystyle\int\limits_{-\infty}^{\infty}\dfrac{x}{\pi(1+x^2)}dx$ does ...
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The mean value of sample moment with order k

I've this problem of statistics that I can't resolve. So I hope that someone can halp me. The problem is this: Let sampling moments $\overline{X_n^k}=\frac{1}{n}\sum_{i=1}^nX_i^k$, where for $k=1$ $...
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AM-GM-HM Relationship

𝑥 and 𝑦 are the geometric mean and the harmonic mean of any two positive (identical or nonidentical) integers. Calculate the minimum value of their arithmetic mean in terms of 𝑥 and 𝑦. HM= $\...
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What is the mean of a random number with one exclusive bound?

If I have a random number between 0 and 1 inclusive, the mean is 0.5. If both are exclusive, the distance from 0.5 to 0 and 0.5 to 1 are the same, so 0.5 is still the mean. But if 0 is inclusive and 1 ...
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Sampling binary values from a discrete probability distribution

Suppose that we have a discrete non-uniform probability distribution X over $\{0,1\}^k$ for n binary noise values. Let $e = (e_1 ,..., e_m)$ be a vector of independant identically distributed binary ...
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Definition of sample mean

I've seen two definitions of sample mean on the internet. One definition defines it as the average of Random variable other defines it as the average of sample values of a sample. I'm confused which ...
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Limit of a recursive sequences involving the AM, GM, and HM (arithmetic-geometric-harmonic mean)

Let $x,y,z$ be positive real numbers. And let $\text{AM}$, $\text{GM}$, $\text{HM}$ respectively be the arithmetic mean, geometric mean, and harmonic mean. Define $$a_n=\text{AM}(a_{n-1},g_{n-1},h_{...
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What is the general term of this series (obtained from an iterated mean)?

This is a long overdue follow up to another question, but it's worth asking separately. Consider the following iterated mean: $$a_{n+1}=\frac{a_n+b_n}{2}$$ $$b_{n+1}=a_n+b_n-\sqrt{a_n b_n}$$ ...
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Finding covariance function

Let $X(t) = A_0 +A_1t+A_2t^2$, where $A_i's$ are uncorrelated random variables with mean $0$ and variance $1$. Find the mean function and covariance function of $X(t)$. I do not have any clue on how ...
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Mean and Variance of ${S ^ 2}$ when $\frac{(n-p)}{ \sigma ^ {2} } \cdot {S ^ 2} \longrightarrow {\chi ^ 2 }{(n-p)}$

I am looking for the detailled proof for this case, if $$\frac{(n-p)}{ \sigma ^ {2} } \cdot {S ^ 2} \longrightarrow {\chi ^ 2 }{(n-p)}$$ Prove that : $$E[{S ^ 2}] = \sigma ^ {2}$$ $$Var[{S ^ 2}] =...
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Prove that $\lim_{k\to\infty}{ \sqrt[n]{ \prod_{i=1}^{n}{(x_i+k)}} - k } = \frac{\sum_{i=1}^{n}{x_i}}{n}$

I accidentally discovered this equality, in which I can prove numerically using python. $$\lim_{k\to\infty}{ \sqrt[n]{ \prod_{i=1}^{n}{(x_i+k)}} - k } = \frac{\sum_{i=1}^{n}{x_i}}{n}$$ But I need ...
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Confusing question on finding mean from frequency tables.

My attempt: The missing frequency = 6 (from 50 games which represents total frequency) Total number of tickets won by Adan = 203 Mean number of tickets won per game = $\frac{203}{50}=4.06$ From ...
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Finding new mean and standard deviation given initial values for both

I'd appreciate any assistance on this: On average, a salesperson makes a sale to 60% of their contacts in a given day with a standard deviation of 2.2%. Assume the salesperson makes 20 contacts per ...
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How to calculate the updated centroids of clustering?

I have 100 points on a two dimentional space. Any point $i$ is defined by the coordinates $(x_i,y_i)$. Lets say we perform Kmeans clustering over these points and generate clusters. Now, each ...
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1answer
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AM GM inequality using conditional extrema

Given: $\prod_{i=1}^n x_i = 1$ leads to constraint function $G(x_1,x_2,...,x_n)=\prod_{i=1}^n x_i-1$ ($\prod_{i=1}^n x_i =x_1 x_2...x_n$) Task is to to find the minimum using conditional extrema of ...
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variance of estimator $\hat{\mu}_1=\bar{X_1}$

From a population with mean $\mu$ and variance $\sigma^2$, an independent random sample of size $n_1$ is extracted. The mean sample is $\bar{X_1}$. The next estimator is proposed: $$\hat{\mu}_1=\...
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Interpret Logarithmic Values like Mean and Std

let's say i have a series of data containing prices on the log scale. How would i interpret a arithmetic mean of 0.55 and a std of 0.69 (both metrics are computed with the log prices). Is there an ...