Variance Definition and 329 Threads

Going through my notes ... and I see the following: 1. Variance = ##\displaystyle \text{Var(X)} = \sigma^2 = \frac{1}{n - 1} \sum_i = 1 ^n \left(x_i - \overline x \right)^2## 2. Standard Deviation = ##\sigma = \sqrt {Var(X)} = \sqrt {\sigma^2}## Both variance and standard deviation are...

My question relates to subsection 2.2.1 of [this article][1]. This subsection recalls the work of Lindgren, Rue, and Lindström (2011) on Gaussian Markov Random Fields (GMRFs). The subsection starts with a two-dimensional regular lattice where the 4 first-order neighbours of $u_{i,j}$ are...

I am trying to find the expected value of the variance of energy in coherent states. But since the lowering and raising operators are non-hermitian and non-commutative, I am not sure if I am doing it right. I'm pretty sure my <H>2 calculation is right, but I'm not sure about <H2> calculation...

1. You are working in a company facing attrition problems of the customer service representatives for the past five years. The company president proposed that if the attrition rate is at least 10 per month, then the salary scale, compensation package, and professional development programs for...

OK, Let me attempt part (i), first, Here we have; ##s^2_p ##=##\dfrac{ (n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}## ##s^2_p ##=##\dfrac{ (7-1)0.63953+(7-1)0.6148}{7+7-2}## ##s^2_p ##=##\dfrac{3.83718+3.6888}{12}## ##s^2_p ##=##\dfrac{3.83718+3.6888}{12}## ##s^2_p ##=##\dfrac{7.52598}{12}##...

I've a Gaussian momentum space wavefunction as ##\phi(p)=\left(\frac{1}{2 \pi \beta^{2}}\right)^{1 / 4} e^{-\left(p-p_{0}\right)^{2} / 4 \beta^{2}}## So that ##|\phi(p)|^{2}=\frac{e^{-\left(p-p_{0}\right)^{2} / 2 \beta^{2}}}{\beta \sqrt{2 \pi}}## Also then ##\psi(x, t)=\frac{1}{\sqrt{2 \pi...

I have a problem of understanding in the following demo : In a cosmology context with 2 probes (spectroscopic and photometric), let notice ##a_{\ell m, s p}## the spectroscopic and ##a_{\ell m, p h}## the photometric coefficients of the decomposition in spherical harmonics of the distributions...

Hi all - I wonder if you can help please. Watching a video on youtube to help me understand about the mean, variance and standard deviation but last part of video left me confused. The speaker said the following for the formula for standard deviation: Consider if the variance is 200 for the...

Just to remind, ##C_\ell## is the variance of random variables ##a_{\ell m}## following a Gaussian PDF (in spherical harmonics of Legendre) : ##C_{\ell}=\left\langle a_{l m}^{2}\right\rangle=\frac{1}{2 \ell+1} \sum_{m=-\ell}^{\ell} a_{\ell m}^{2}=\operatorname{Var}\left(a_{l m}\right)## 1)...

It is in cosmology context but actually, but it is also a mathematics/statistical issue. From spherical harmonics with Legendre deccomposition, I have the following definition of the standard deviation of a ##C_\ell## noised with a Poisson Noise ##N_p## : ## \begin{equation}...

The question is below: below is my own working; the mark scheme for the question is below here; i am seeking for any other approach that may be there...am now trying to refresh on stats...bingo!

I have the following expression for an error on a Cℓ : ##\sigma(C_{\ell,X})=\sqrt{\dfrac{2}{(2 \ell+1)\Delta\ell \,f_{\mathrm{sky}}}}\left[C_{\ell,X}+N_{X}(\ell)\right]## where ##X## corresponds to spectroscopic/photometric shot noise and with ##\Delta\ell## is the bin width between 2 values...

This is not actually a homework problem. I'm old but having trouble with something that's probably at student level because it's so long since I learned this stuff. I would be grateful if someone would please take pity on me and help me out! I am trying to calculate something that includes...

Hi, I was attempting the following problem and I didn’t know how to start it off correctly. Question: At a party there are ##n## couples. When the last song comes on, each person randomly picks a dance partner. What is the: (a) mean, (b) variance of the number of couples that are paired...

Hello, In the context of Legendre expansion with ##C_\ell## quantities, below the following formula which is the error on a ##C_\ell## : ##\sigma_(C_{\ell})=\sqrt{\frac{2}{(2 \ell+1)\Delta\ell}}\,C_{\ell}\quad(1)## where ##\Delta\ell## is the width of the multipoles bins used when computing...

$\newcommand{\szdp}[1]{\!\left(#1\right)} \newcommand{\szdb}[1]{\!\left[#1\right]}$ Problem Statement: Let $S_1^2$ and $S_2^2$ denote, respectively, the variances of independent random samples of sizes $n$ and $m$ selected from normal distributions with means $\mu_1$ and $\mu_2$ and common...

Hi, I was looking at this problem and just having a go at it. Question: Let us randomly generate points ##(x,y)## on the circumference of a circle (two dimensions). (a) What is ##\text{Var}(x)##? (b) What if you randomly generate points on the surface of a sphere instead? Attempt: In terms of...

There is a tall cylinder filled with water. And there is a 3 in diameter hole near the bottom and water is gushing out. (assume the cylinder is continually being re-filled from the top) You work to plug the hole with a 10 inch long cylinder that is exactly the perfect diameter fit to plug the...

This is a question from a mathematical statistics textbook, used at the first and most basic mathematical statistics course for undergraduate students. This exercise follows the chapter on nonparametric inference. An attempt at a solution is given. Any help is appreciated. Exercise: Suppose...

I have an expression of Matter Angular power spectrum which can be computed numerically by a simple rectangular integration method (see below). I make appear in this expression the spectroscopic bias ##b_{s p}^{2}## and the Cosmic variance ##N^{C}##. ## \begin{aligned}...

In my book, when calculating the variance of X = (x_1 + x_2 + x_3 + x_4 + x_5)/5 in an example it says: V(X) = V(1/5(X_1 + X_2 + X_3 + X_4 + X_5)) = 1/25*V(X_1) + 1/25*V(X_2) + 1/25*V(X_3) + 1/25*V(X_4) + 1/25*V(X_5) = 1/5Ф I don't understand how V(1/5X) can be turned into 1/25*V(X), shouldn't...

Hello, I would like to know the right expression for the expression of variance of Shot noise in spectroscopic probe. Sometimes, I saw ##\sigma_{SN,sp}^{2} = 1/n_{sp}## with ##n_{sp}## the average density of galaxies, whereas my tutor tells me that ##\sigma_{SN,sp}^{2} = 1/n_{sp}^{2}## , so I...

I cite an original report of a colleague : 1) I can't manage to proove that the statistical error is formulated like : ##\dfrac{\sigma (P (k))}{P(k)} = \sqrt{\dfrac {2}{N_{k} -1}}_{\text{with}} N_{k} \approx 4\pi \left(\dfrac{k}{dk}\right)^{2}## and why it is considered like a relative error ...

Below the error on photometric galaxy clustering under the form of covariance : $$ \Delta C_{i j}^{A B}(\ell)=\sqrt{\frac{2}{(2 \ell+1) f_{\mathrm{sky}} \Delta \ell}}\left[C_{i j}^{A B}(\ell)+N_{i j}^{A B}(\ell)\right] $$ where ##_{\text {sky }}## is the fraction of surveyed sky and ##A, B##...

Hey! :giggle: Let $X$, $Y$ and $Z$ be independent random variables. Let $X$ be Bernoulli distributed on $\{0,1\}$ with success parameter $p_0$ and let $Y$ be Poisson distributed with parameter $\lambda$ and let $Z$ be Poisson distributed with parameter $\mu$. (a) Calculate the distribution...

There are 8 items in a data set. The sum of the items is 48. a. Find the mean. $\qquad\textit{mean}=\dfrac{\textit{sum}}{\textit{data set}}=\dfrac{48}{8}=\textbf{6}$ The variance of this data set is 2. Each value in the set is multiplied by 3. b. Write down the value of the new mean. $\qquad...

I found the mean to be $$\langle n\rangle=\vert\alpha\vert^2 \tanh(\alpha^2)$ and $\langle n^2\rangle=\vert\alpha\vert^2 \left( \alpha^2\sech(\alpha^2)^2 + \tanh(\alpha^2) \right)$$. Do you know if there is any reference where I can check if this is correct?

Hello everyone. I have two points in space (on the surface of the earth) represented using spherical coordinates (in this case there is no z axis since both are assumed to be at the same height). These points have an associated standard deviation in lambda and in phi, which are longitude and...

Hello! (Wave) Suppose that we calculate the calories and the quantity of sugar at the package of a product. For the calories we have mean value $10$ and standard variation $4,90$. For the quantity of sugar we have corresponding values $5,85$ and $3,38$, respectively. (Use CV). I want to find...

I derived (trivially) an expression for the variance of a random variable (which I had never noticed before). Let ##X## be a random variable with cdf ##F(x)## then (assuming finite second moment). ##Var(X)=\frac{1}{2}\int\int (x-y)^2dF(x)dF(y)##. Is this expression of any use?

In a paper published in the JOURNAL OF MATHEMATICAL PSYCHOLOGY 39, 265-274 (1995), formulas are provided on page 272 for the expectation E(Tn) of a random variable T as dependent on n (formulas 28 and 29). Now I would like to know what these formulas look like for the variance i.e. Var(Tn).

At first I assumed u to be 1/2 since X is equally distributed along 0-1. $$\int (x-1/2)^2*x^2 dx = 1/30$$ The correct answer should be 4/45. I would calculate the u but I think I do it wrong. If fx(x) = x^2 then what is g(x)? fx is the probability density function, which is the x^2 they...

We have a sample of X, a Normalized Gaussian random variable.We divide the data into positive and negative. Each will have a conditional variance of ## 1−\frac{2}{π}## . Can someone show how to get this result ? I found this problem here (page 3) ...

I found that <x> of p(x) = 1/π(x2 + 1) is 0. But its <x^2> diverges. I don't know if there are other ways of interpreting it besides saying that the variance is infinity. I usually don't see variance being infinity, so I'm not sure if my answer is correct. So, can variance be infinity? And does...

THE PROBLEM 6) Table 1 below contains data on offensive statistics for each game in the 2019 UW Husky Baseball season. Answer the following questions and/or complete the specified tasks using these data. Do everything by hand and show your work (good practice for the tests). a. Construct a...

Given the probability density function f(x) = b[1-(4x/10-6/10)^2] for 1.5 < x <4. and f(x) = 0 elsewhere. 1. What is the value of b such that f(x) becomes a valid density function 2. What is the cumulative distribution function F(x) of f(x) 3. What is the Expectation of X, E[X] 4. What is...

Hello everyone. I am currently using the pca function from MATLAB on a gaussian process. Matlab's pca offers three results. Coeff, Score and Latent. Latent are the eigenvalues of the covariance matrix, Coeff are the eigenvectors of said matrix and Score are the representation of the original...

Hi everyone in this link (https://stats.stackexchange.com/questions/226334/ljung-box-finite-sample-adjustments) I see the variance of autocorrelation related to specific lag is demonstrated in the following: $$ Var(r_k) = \frac {\sum_{t=k+1}^n a_t*a_{t-k}} {\sum_{t=1}^n a_t^2}$$ where ##r_k## is...

Hey, I've got this problem that I've been trying to crack for a while. I can't find any info for multi-variable expected values in my textbook, and I couldn't find a lot of stuff that made sense to me online. Here's the problem. Find $E(C)$ Find $Var(C)$ I tried to get the limits from the...

Homework Statement I have simulated Langevin equation (numerically in Matlab) for some specific conditions, so I have obtained the solution ##X(t)##. But now, with the solution I have obtained, I have to calculate ## <X(t)|x_0>, <X^2(t)|x_0>-(<X(t)|x_o>)^2 ## and the conditional correlation...

A coin had tossed three times. Let ##X##- number of tails and ##Y##- number of heads. Find the expected value and variance ##Z=XY##. My solution: We know, that ##Y=3-X##, so ##Z=(3-X)X## for ##X=0,1,2,3##. ##Z=2## for ##X=1,2## and ##Z=0## for ##X=3,0## So, ##E(Z)=E((3-X)X))= 2 \cdot ⅜ +2 \cdot...

Homework Statement [/B] For reference: Book: Mathematical Statistics with Applications, 7th Ed., by Wackerly, Mendenhall, and Scheaffer. Problem: 10.81 From two normal populations with respective variances ##\sigma_1^2## and ##\sigma_2^2##, we observe independent sample variances ##S_1^2## and...

Problem: We play roulette in a casino. We watch 100 rounds that result in a number between 1 and 36. and count the number of rounds for which the result is odd. assuming that the roulette is fair, calculate the mean and deviation Solution: I understand that the probability - Pr = 0.5. and...

Homework Statement For a wavefunction ##\psi##, the variance of the Hamiltonian operator ##\hat{H}## is defined as: $$\sigma^2 = \big \langle \psi \mid (\hat{H} - \langle\hat{H}\rangle)^2 \psi \big\rangle$$ I want to prove that if ##\sigma^2 = 0##, then ##\psi## is a solution to the...

Homework Statement Random variable Y has a binomial distribution with n trials and success probability X, where n is a given constant and X is a random variable with uniform (0,1) distribution. What is Var[Y]? Homework Equations E[Y] = np Var(Y) = np(1-p) for variance of a binomial...

Hi, Let y = x + z, where x and z are mutually independent RVs. Also, z is a complex gaussian RV with zero mean and variance sigma^2. My question is as follows: For x = y - z, what is the variance of (-z) ? Any help could be useful. Thanks in advance.

<Moderator's note: Moved from a technical forum and thus no template.> So, I have this problem and I am stuck on a sum. The problem I was given is the following: The probability of a given number n of events (0 ≤ n < ∞) in a counting experiment per time (e.g. radioactive decay events per...

Suppose ##X## ~ N(##\mu##,##\sigma^2##). Then ##\bar{X}## ~ N(##\mu##,##\frac{\sigma^2}{n}##), where ##\bar{X}## is the random variable for sample mean for samples of size ##n##. But when the population variance ##\sigma^2## is unknown and the sample size ##n## is small, ##\bar{X}## no longer...

Homework Statement [/B] ##-1\leq\alpha\leq 1## ##f(y_1,y_2)=[1-\alpha\{(1-2e^{-y_1})(1-2e^{-y_2})\}]e^{-y_1-y_2}, 0\leq y_1, 0\leq y_2## and ##0## otherwise. Find ##V(Y_1-Y_2)##. Within what limits would you expect ##Y_1-Y_2## to fall? Homework Equations N/A The Attempt at a Solution...

Homework Statement A particle is moving in a one-dimensional harmonic oscillator, described by the Hamilton operator: H = \hbar \omega (a_+ a_- + \frac{1}{2}) at t = 0 we have \Psi(x,0) = \frac{1}{\sqrt{2}}(\psi_0(x)+i\psi_1(x)) Find the expectation value and variance of harmonic oscillator...