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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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 '...
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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 ...
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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 ...
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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?
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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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Mean Distance Integration
Suppose I parametrize the 2-sphere as
\begin{align}
\vec{r}(u,v)=
\begin{bmatrix}
\cos u\sin v\\
\sin u\sin v\\
\cos v
\end{bmatrix}\ ,\ (u,v)\in[0,2\pi]\times[0,\pi]
\end{align}
and I have a curve ...
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Can some measure like mean or median be "proved" to be a good indicator of something in a situation?
in statistics, can some measure like middle or median be kind of "proved" to be a good indicator of something in a situation, if you cannot come up with a counterexample which clearly shows ...
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Poisson Distribution Mean and Variance
I know that the mean and variance of a Poisson distribution is λ, so I don't understand how the mean and variance in the question are 70 and 700 respectively. Here's question for reference.
I am the ...
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Relationship between mean of square and mean of cube to square and cube
Say I have a quantity I am measuring in time $u=\bar{u}+u'$, where $\bar{u}$ is the mean value and $u'$ is the fluctuating value. How can I find the relationship between $(\bar{u})^2$, $(\bar{u})^3$ ...
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The "benchmark" problem - finding a comparable "mean" over variables of unknown mean
I have a tuple of variables $$(X_1, ..., X_n)$$ that are a sample of what I call a "benchmark test", eg. each variable represent a value that depends on the performance of a system, but ...
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Show that $\sqrt{4x^2+12xy+5y^2}+\sqrt{4y^2+12yz+5z^2}+\sqrt{4z^2+12xz+5x^2}<=13\sqrt{21}$
question
Let the real numbers $x,y,z$ be positive nonzeros that fulfil the condition $x+y+z=13$. Show that $\sqrt{4x^2+12xy+5y^2}+\sqrt{4y^2+12yz+5z^2}+\sqrt{4z^2+12xz+5x^2}<=13\sqrt{21}$.
idea
...
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Let $x$, $y$, and $z$ be positive real numbers. Prove the following inequality...
Question
Let $x$, $y$, and $z$ be positive real numbers. Prove the inequality:
A. $\frac{x^4+x^2+7}{y+2}+\frac{y^4+y^2+7}{z+2}+\frac{z^4+z^2+ 7}{x+2}>=x+y+z+6$
B.
$\frac{x^4+x^2+7}{(y+2)^2}+\frac{...
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Estimating Quantity $ p _2$ from $ p _1 $ with Mean Error Correction
If $p_1$ and $p_2$ are two quantities, where $P_1$ and $P_2$ are random variables representing them, and if the mean of the error is $\mu = E(P_1 - P_2)$, can the values of $p_2$ be estimated as $p_2 =...
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Help Analyze whether, at a given moment, the following cases are possible....
Question
The numbers $4$, $8$, $9$ and $15$ are written on the board. Carla deletes three of them and then writes three more numbers following the rule: if the numbers $a$, $b$, $c$ are deleted, the ...
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Analyzing the Impact of Mean Subtraction on Series
I have two random series, $X$ and $Y$, and I've defined a third series $Z$ as:
$Z = X - Y$
Now, I want to compare two other series, $Y1$ and $Y2$, to understand why $Y1$ is generally closer to $Y$ ...
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Why is the average of $\sin^2 wt = 1/2$ and $\cos^2 wt = 1/2$ [closed]
Why is the average of both $\sin^2 = \frac{1}{2}$ and $\cos^2 = \frac{1}{2}$
I was revising Simple Harmonic motion notes and in the average of Kinetic energy derivation
$$KE = \frac12 k A^2 \cos^2(\...
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Diffrence in probability distributions of sepertaed groups [closed]
If I were to measure some quantitavie metric of a sample population and record its mean, and then I were to split by random selection all members of the population into two groups of equal size and ...
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Average side length of a triangle with perimeter $p$
On the one hand, I think that by symmetry the average side of a triangle with given perimeter $p$ is $\frac{p}{3}$.
However (and here I'm probably mistaken), if I look at a side of the triangle, say $...
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How to calculate the average stock price (level)
I know that we can calculate average stock price by using mean but this doesn't give a real picture of the portfolio specially if we have bought and sold the same stocks.
How do we calculate average ...
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Are there any tighter bounds on the mean number of orthogonally connected components in a binary random rectangular grid?
Question 17 of the set of 20 probability problems at https://www.math.ucdavis.edu/~gravner/MAT135A/resources/chpr.pdf provides bounds of $\tfrac{mn}{8}$ and $\tfrac{(m+2)(n+2)}{6}$ for the mean number ...
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Asymptotic scaling of mean and variance for non centered chi distribution
Defined $Y \equiv \sqrt{\sum_{i=1}^k(\frac{ X_i}{\sigma_i})^2}$, with $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$ it is known that Y is distributed as a non-central chi (Noncentral chi distribution); ...
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The weighted average of weights will be smaller than will be smaller than the weighted mean in some cases. [closed]
Suppose I have $w_i>0$ and $x_i>0$ for $i=1,2,\ldots, n$. I know that,
$$\sum_{i} w_i < \sum_{i} x_i \leq 1. \tag{1} \label{eq1}$$
Can I say that,
$$\sum_{i} w_i w_i < \sum_{i} w_i x_i. \...
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Collisions in random functions
Consider a random function, from $\{0, 1\}^n$ to $\{0, 1\}^n$.
For a particular string $y^{*} \in \{0, 1\}^n$ in the image of the function, in expectation over the randomness, how many strings in the ...
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Derivative of mean with respect to weights leads to mean always increasing?
I'm having this counterintuitive result that hopefully you fellas can shed me some light.
Imagine a discrete p.d.f with n classes. The mean of this distribution can be written as:
$$ E[X] = \sum_{i=1}^...
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How to add new rows into a matrix $X$ without change its center?
Assume that you have a matrix $X \in \Re^{m \times n}$ and you have centered it column wise.
$$\mu_j = \frac{1}{m}\sum_{i=1}^{m} X_{i,j}$$
$$X = X - \mu_j$$
MATLAB code:
...
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Minimizing the variance of a linear combination of random variables
Let $u, v, w, x, y$ be vectors in $\mathbb{R}^m$ where each component is an observation of a random variable $U, V, W, X, Y$. Let $\bar{u}, \bar{v}, \bar{w}, \bar{x}, \bar{y}$ be the average of $u, v, ...
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Proving $\overline{(x-\overline x)(y-\overline y)}=\overline{xy}-\overline x \cdot \overline y$
In ordinary least squares, it can be shown that $\hat\beta_1 = \frac{\sum(x_i-\bar x)(y_i-\bar y)}{\sum (x_i - \bar x)^2}$. I'm trying to prove this. I arrived at $$\hat\beta_1=\frac{\overline{xy}-\...
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Mean of Medians? Median of Means?
This is a question I have wondered about for a long time and have never been able to find a full mathematical explanation behind this.
Suppose there are 100 countries. As an experiment:
We give ...
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Expected Number of Dice Rolls to See All Sides
I want to check if the solution to the following question is an application of the property $E[X + Y] = E[X] + E[Y]$.
The question:
What is the expected number of rolls needed to see all six sides of ...
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Can we find the probability using the mean and median ? If it is possible, how can we calculate it?
Lucas is a student who did a math competition against 2875 other people. His rank in his results will allow him to have a school. Everybody has distinct ranks. The nearest of 1 is better. The result ...
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What is the range of values after applying Standardization?
I was working on a ML model, and after applying Feature Scaling (Standardization), I observed some of the scaled value is lesser than -3 and some are greater than +3.
...
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Conjecture about the mean value of an almost periodic function (the "Mountains of Guilin")
Consider the function $f_{p_j,n}(x)=|\sin (p_1x)+\sin (p_2x)+\dots+\sin (p_nx)|$ where $p_j$ is any sequence such that no two $p$ are rational multiples of each other where the $p_j$ are linearly ...
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Consistency of complete-case analysis or mixed random variable mean estimator
Let $Y$ be a random variable and $R \in \{0,1\}$ a random variable that indicates whether $Y$ is observed or missing
\begin{equation}
Y \cdot R = \begin{cases}
Y & R=1, \\
0 & R=0.
\end{cases}
...
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mean median and maximum product question
The mean and the median of nine distinct positive integers is 30. If the nine integers are such that their product is maximum, what is the product of the smallest and the largest integers?
My attempt:
...
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Can you partition the set of 9 consecutive integers 1 to 9 in 2 sets, s.t. no member of either set is the mean of two other members of the same set? [duplicate]
Is it possible to partition the set $\Omega=\{1,2,3,4,5,6,7,8,9\}$ in two subsets $\Omega=A\cup B$, $A\cap B=\emptyset$, such that no member of either subset is the mean of two other members of the ...
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If AM-GM fails for some list of $n_0\ge 2$ numbers, then for all $n\ge n_0$, there's a list of $n$ numbers such that AM-GM fails
the exercise
I want to prove this by induction, but I'm stuck.
In the induction step, I assume that there's a list of $n$ numbers such that the inequality fails, and then I should find a list of $n+1$ ...
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Enhancing the precision of a mean of cosine/sine computation
I wrote a program that computes incrementally the mean and variance of values generated with the function $a\cos(\omega t)+\mu$.
The program will have to compute these for sums of cosine and sine of ...
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Mean distance vs distance of means
I have a series of (x,y) coordinate pairs representing observations around two fixed points with random error in each observation. The random error is compensated for by taking the average of a large ...
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Find all whole number $k$ with the property: $k$ is equal to the arithmetic mean of all the numbers obtained from all the possible permutation of $k$.
The exact question is here
A whole number is equal to the arithmetic mean of all the numbers obtained from the given number with the aid of all possible permutation of its digits. Find all whole ...
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Is it legit to compute the mean of log values? [closed]
I have two vectors where each element represents a value log2 transformed.
v1 = c(1.4, 2.1, 1.9)
v2 = c(-1.2, -2.2, -1.9)
I'd like to compute the mean of v1 and v2 ...
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When does $f(x) = avg\int_{B(x,r)}{f(z)dz} $ imply $\Delta f = 0$?
Suppose $f \; : \; \mathbb{R}^n \to \mathbb{R}$ is an $\mathbb{L}^{1}_{loc}$ function (defined everywhere!) $\mathcal{R}\subset(0,+\infty)$ is a set such that (*) holds, meaning :
$$f(x) = avg\int_{B(...
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I have read the statement Expectation of two independent variable X and Y E(XY)=0. How to prove it?
I know that if X and Y are independent variables then the expectation value of X and Y i.e $ E(XY)=E(X).E(Y).$ But then the expectation value will be equal to zero only when either of them i.e $E(X) ...
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Find the average value of the function on the segment
The function is given:
$$\chi(\xi) = - Re \left[\sqrt[3]{\sqrt{\displaystyle P^3 - 3 \cdot \xi^2 \cdot P^2 + 3 \cdot \xi^4 \cdot P} + \xi^3-\xi} \cdot (1+ j \cdot \sqrt{3}) \right].$$
it is necessary ...
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Find expected value of product of wiener process and stochastic integral
Find expected value of $$Y_t = W_{1/4}\int_{0}^{1/2}\cos{t}\,dW_t$$
Firstly, I thought that $E[Y_t] = 0$. But then I realized that
$ W_{1/4}\int_{0}^{1/4}\cos{t}\,dW_t $ could be dependent.
We can ...
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Statistical indexes order in an asymmetrical distribution
I was wondering what's the order between mode, median, arithmetic mean, geometric mean and harmonic mean in a distribution with positive skewness and in another with negative skewness. Which are the ...
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Can anyone help me to calculate this finite sum?
Problem:
Let $f(x)=\frac{1}{2\epsilon}\chi_{[\frac{1}{2}-\epsilon,\frac{1}{2}+\epsilon]}(x)$ where $\epsilon>0$. Calculate $$\frac{1}{n}\sum_{k=0}^{n-1}f(x+k\alpha)$$ on $\mathbb{T}=[0,1]/\sim$.
...
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Having an empirical chart, what is the probability that the mean of a 99,000 sized set of bytes is $\le 127$?
A chart of means of sets of pseudo random numbers:-
The x axis is the size of the random set $X$, again chosen randomly between 20,000 and 1,024,000. The y axis is the calculated mean of that set. ...
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Unit weighted average
Suppose that I have the following sample
$$\text{Farmer}_{\text{ID}}$$
$$\text{Farmer}_{\text{Size}}$$
$$\text{Range}$$
$1$
$27$
$a) \leq 100$
$2$
$82$
$a) \leq 100$
$3$
$91$
$a) \leq 100$
$4$
$...
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How does one find the expected value of the Conway base-13 function $f:\mathbb{R}\to\mathbb{R}$, when restricted to $f:[0,1]\to[0,1]$?
Main Question:
Using the Lebesgue measure, what is the expected value of the Conway base-13 function $f:\mathbb{R}\to\mathbb{R}$, when restricted to $f:[0,1]\to[0,1]$?
Attempt:
I’m not sure what the ...
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