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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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Modified AGM: $a_{n+1}=\frac{a_n+b_n}{2}, \quad b_{n+1}=a_n+b_n-\sqrt{a_n b_n}$

The idea is as follows: generate a sequence from two numbers by subtracting their means from their sum, for example arithmetic and geometric means: $$a_{n+1}=a_n+b_n-\frac{a_n+b_n}{2}=\frac{a_n+...
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Limit and rate of convergence of the sequence $a_{n+1}=\frac{a_n^2+b_n^2}{a_n+b_n},~~b_{n+1}=\frac{a_n+b_n}{2}$

Define the sequence the following way for some $x,y \geq 0$: $$a_0=x,~~~~~~~b_0=y$$ $$a_{n+1}=\frac{a_n^2+b_n^2}{a_n+b_n},~~~~~~b_{n+1}=\frac{a_n+b_n}{2}$$ Obviously: $$a_n \geq b_n,~~~~n \geq 1$$ ...
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Prove with some AM-GM inequality?

I have proved the following inequality: Let $a,b,c>0$ $$\dfrac{(a+\sqrt{ab}+\sqrt[3]{abc})}{3}\le \sqrt[3]{a\cdot\dfrac{a+b}{2}\cdot\dfrac{a+b+c}{3}}$$ My solution is:$$a\dfrac{a+b}{2}\cdot\dfrac{...
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Mean Value and Variance of a Birth and Death Process

Let $\{X(t)\}_{t>0}$ on $\{0,1,2,3\}$ a birth and death process, with $\lambda(s)=(3-s)^2$ and $\mu(s)=s^2+s$. Assume $P(X(0)=3)=1$ and determine: (a)$E[X(t)]$; (b)$Var[X(t)]$. I don't know how ...
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Does this three number mean have a name? (Carlson elliptic integrals)

Recently I found out about Carlson elliptic integrals, which have great symmetry properties and allow to compute every kind of elliptic integrals and other functions. The question is about the method ...
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how to subtract mean from a set of SPD matrices

I have a set of SPD matrices and I know how to calculate their mean. My question is: Is there any method to subtract the mean from each sample? In Euclidean space we simply subtract the mean from ...
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Arithmetic-quadratic mean and other “means by limits of means”

For $x,y$ positive real numbers, and $p\neq 0$ real, define the Hölder $p$-mean $$M_p(x,y) := \left(\frac{x^p+y^p}{2}\right)^{1/p}$$ whereas $$M_0(x,y) := \sqrt{xy}$$ is the limit of $M_p(x,y)$ when $...
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Mean value of stochastic process with random variable as the index

Let $\{X_t: t\geq 0\}$ be a stochastic process on the probability space $(\Omega,\mathcal{A},\mathbb{P})$ with values in $\mathbb{R}$ and $T:(\Omega,\mathcal{A},\mathbb{P})\rightarrow\mathbb{R}^{+}_0$ ...
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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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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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Is this rule a consistent estimator?

Define the sample mean $S_n=(X_1+...+X_n)/n$ of a sequence $\{X_1,...,X_n,...\}$ of i.i.d. random variables with non-negative discrete support, known mean $0<\mu<1$, finite standard deviation $\...
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Exclude worst value from weighted arithmetic mean

I'm working on a tournament rating, that is calculated as a weighted arithmetic mean, i.e. the formula is: $$ R = \frac{a_1 x_1+a_2 x_2+\cdots+a_n x_n}{a_1+a_2+\cdots+a_n} $$ $a_i$ is a positive ...
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Conditional convergence of $\Bbb E[X^{-1}]$ for $X\sim\mathcal{N}(\mu,\sigma^{2})$ as $\operatorname{pr}(X>0)\to 1$

This question references the post Reciprocal of a normal variable with non-zero mean and small variance In the question the OP simulated observations from an inverse normal distribution, i.e. ...
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Poisson Sample Mean Wald Test

Let $X_1,\ldots,X_n\sim\text{Poisson}(\lambda)$, $H_0:\lambda=\lambda_0$, and $H_1:\lambda\neq\lambda_0$ for $\lambda_0>0$. Compute the size $\alpha$ Wald test, estimating $\lambda$ by $\overline{X}...
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Mean of a Poisson process

I have $$X(t), -\infty \leq t \leq \infty $$ as a Poisson process with parameter $ \lambda $ and then $$Y(t) = X(t)-tX(1), 0 \leq t \leq 1$$ When I find the mean of $Y(t)$, I simply do $ E(Y(t)) = E(...
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Hint for computing the mean and variance of $\hat{\alpha} = - \frac{n}{\log \prod_{i=1}^n X_i }$

In my statistical class, we are learning MLE. Last week we end up computing the MLE of parameter $\alpha$ for a random sample $X_1, X_2,...,X_n$ with $\text{Beta}(\alpha,1)$ distribution with support $...
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If the diagonal of a rectangular box measures L, then what is its maximum volume?

If the diagonal of a rectangular box measures L, then what is its maximum volume ( in terms of L ) ? $L = \sqrt{x^2 + y^2 + z^2}$, then $L^2 = x^2 + y^2 + z^2$. By AM-GM, $$L^2 \ge 3V^{\frac{2}{3}}$...
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Limit superior and inferior of Cesàro means

From a sequence $a^x_n$, define the sequence of its Cesàro means $a^{x+1}_n$ as $$\sum_{k=1}^{n} a^x_k/n$$ It is easy to show that the sequence of Cesàro means of a bounded sequence will itself be ...
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Is the mean of the Weierstrass function defined?

The Weierstrass function is an example of a function that is continuous everywhere but differentiable nowhere. My question is whether the mean $\bar{f}$ of the Weierstrass function $f(x)$ can be ...
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game with 3 dice - profit of player

A player rolls 3 dice (6 sides) simultanuously. If at least one 5 or 6 appeared he wins 2 dollars. In any other case he loses 6 dollars. a) Find pdf of random variable $Y$ profit of player at a game b)...
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The expected value as an integral

Let $X \geq 0$ be continuous random variable with the CDF $F(x) := \displaystyle\int_0^x f(x) \; \mathrm dx$ where $f(x)$ is the PDF. I want to express the (finite) expected value $E[X] := \...
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Mean Value of a Markov Process

I wonder if there is a general approach to computing the mean value $\mu(t)$ of a Markov process without going through the computation of the full probability distribution. I remember (vaguely) that ...
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Is it possible to derive the equation for the arithmetic mean?

As I understand it, the arithmetic mean is a measure of central tendency, i.e. it is a value that quantifies the location of the centre of a distribution of data points (the point about which the data ...
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Mean and variance of a Random variable

I am trying problem no. 2 from Purdue HW here . Balls are drawn from an urn containing $w$ white balls and $b$ black balls until a white ball appears. Find the mean value $m$ and variance $σ^2$ of ...
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Means and Covariances of powers of a normal distribution

Let $X$ be a normally distributed random variable, with mean $\mu$ and variance $\sigma^2$. Consider a random vector $$V = \left[ X^n, X^{n-1}, \dots, X^2, X, 1 \right]^T $$ What is the expected ...
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Arithmetic geometric mean - irrational, algebraic, trancendental?

Are there some general theorems about rationality/irrationality and abgebraicity/transcedentality of arithmetic-geometric mean? At least for some group of numbers (like natural numbers)? Or even for ...
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Upper bound for probability

Given two i.i.d. random variables $X,Y$ that satisfy the following condition: $P(|X|<0.5)<a$ and $P(|Y|<0.5)<b$ How can I derive an upper bound for the following probability $$P\left(\...
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Mean and variance on a metric space

It is my first post, so please correct me if I am not following the rules/etiquette. Assume that we are given a space $\mathcal{S}$ composed by vectors $x\in\mathbb{R}^L$, constrained by $$\sum x_i=...
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Derivative of cumulative generating function at zero equals expectation value

Let $X$ be a random variable with values in $\mathbb{N_0}$. Then we can define the cumulative generating function of $X$ via $$ F_{X}: (-\infty, 0] \rightarrow \mathbb{R} \quad \quad t \mapsto \log(\...
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Average of sum and Sum of average of 2 random non zero distributions of numbers

Suppose we have $$A_i \sim N( \mu_a, \sigma_a)$$ $$B_i \sim N( \mu_b, \sigma_b)$$ Where $A_i$ and $B_i$ are i.i.d. respectively, where $i = 1 \ldots n$, We are interested in $$\frac{ \sum_{i=1}^{n}...
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Asymptotic behavior of truncated mean of zero mean variable

How do you show that $$\sqrt{n}\cdot\mathbb{E}\left[X\cdot \mathbb{1}_{|X|\leq\sqrt{n}}\right]\to 0$$ for a random variable $X$ with zero mean? Do we need finite variance?
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Geometric, Arithmetic, and Harmonic

I'm curious as to the origin of the words "geometric", "arithmetic", and "harmonic" means. What's so "geometric" about the geometric mean? How is the arithmetic mean more "arithmetic" than the other ...
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Riemman-Stieltjes Integration-IVT

I need to show this, Suppose that $f,g: [a,b] \rightarrow \mathbb{R}$ are continous. Show that exist $\eta \in (a,b)$ such that $$g(\eta)\int_a^\eta f(x)dx=f(\eta)\int_\eta^b g(x)dx $$ I defined $...
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MSE Minimized by Mean

In the Gaussian case, it is well-known that the MSE, is minimizer by the mean value. However, in general, if $X \in L^2(\mathcal{F};\mathbb{P})$, is a random-variable in $\mathbb{R}$, then is the ...
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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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Generalized Mean Inequality with STD

I am looking for a formal proof shows that for any $x\&y \geq 0$ and $\alpha > 1$, \begin{equation} \frac{1}{n}\sum_{i=1}^{n}x_i ^\alpha \geq \frac{1}{n}\sum_{i=1}^{n}y_i ^\alpha \end{equation}...
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Mean and correlation of product of two random processes

I have two random process: $$A(at)$$ $$cos(2\pi f_0t+\Phi)$$ with these hypothesis: $a$ and $f_0$ are constant $\Phi$ is uniformly distributed in $[0,\pi)$ $A(at)$ is WSS I must calculate the ...
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Partitioning an equation into summable components

For this actual issue I think the physical setup can most easily be thought of as a tray upon which a set of non-overlapping colored disks are randomly placed. All disks of the same color have the ...
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Calculating discrete mean

I am not sure if a distinction "discrete" and "continuous" mean exists, but it's the best way to describe my problem: I have solved a partial differential equation $f(u^{'}(t),u(t),t)=c$ on a domain $\...
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Mean square continuous Process

Show that a stochastic process $X(t)$ is mean square continuous if and only if its autocorrelation function $R_X(t_1,t_2)$ is continous $\Rightarrow$ Proof: We have $E[(X(t)-X(t_0))^2]=R_X(t,t)-R_X(...
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Poisson and Negative Binomial distributions- Mean and Variance total claim size

I am trying to do the following: Let $S$ be the total claim size when the number of claims follow a Negative Binomial Distribution. How can I derive a formula for the $E(S)$ - expectancy and $V(S)$ -...
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Mean of minimum of an inverse Rician random variable

What is $E\left[\min\left(\frac{1}{X}\right)\right]=?$, where $X$ is a Rician distributed random variable. I know from this article that the first moment of an inverse Rician distributed random ...
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146 views

Nice mean for negative Numbers

I'm searching for a nice mean (or average) for negative numbers that gives less weight to high (absolut) outliers (I don't have enough values to use median). Obviously the way to go for positive ...
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how to estimate weight for weighted product?

I know that one way to estimate the weighted sum method is to use the inverse of each attribute's variance: $1/\sigma^2$. In wiki, it is saying: The significance of this choice is that this weighted ...
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Representing gaussian variables by real scalars

Sometimes we may need to do qualitative analysis of multiple Gaussian variables $\mathcal{N}(\mu_1,\sigma_1^2),\mathcal{N}(\mu_2,\sigma_2^2),\ldots$. A naive strategy would be to just compare their ...
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Deriving confidence intervals for a random variable that is not observed directly (example)

So I have a question about combining random variables. While it is given that a bottle filling machine fills with $\sigma=5$, the observed variable here is the weight of $n$ bottles, where the bottle ...
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Probability problem - Standard Deviation related

Exercise : A car insurance company has $10.000$ customers. The average annual compensation per customer is $240$ with Standard Deviation $800$. Calculate by approximation the probability of the total ...
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Estimate the median from mean, variance etc.

I'm given four groups containing 15 observations each. I can only see a few of the observations. However, for each group, I'm given the variance, mean, sum, USS, SSD etc. I have to estimate the ...
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Variance of the gain, given a set of wagers on $n$ Bernoulli trials

Let's assume we want to wager on a given set of n Bernoulli trials with probabilities $p_i$ with $i \in \{1, 2, ..., n\}$ and a corresponding payoff odds $b_i$, i.e., betting a certain percentage $f_i ...
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Geometric mean of prime twin gaps?

This question is an analogue of Geometric mean of prime gaps? Where primes have been replaced by prime twins. Eric's answer : In 1976 Gallagher proved, under the assumption of a uniform version of ...