# variance Definition and Topics - 18 Discussions

In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its mean. In other words, it measures how far a set of numbers is spread out from their average value. Variance has a central role in statistics, where some ideas that use it include descriptive statistics, statistical inference, hypothesis testing, goodness of fit, and Monte Carlo sampling. Variance is an important tool in the sciences, where statistical analysis of data is common. The variance is the square of the standard deviation, the second central moment of a distribution, and the covariance of the random variable with itself, and it is often represented by

σ

2

{\displaystyle \sigma ^{2}}
,

s

2

{\displaystyle s^{2}}
, or

Var

(
X
)

{\displaystyle \operatorname {Var} (X)}
.

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1. ### A Expression of the mean and variance of spectroscopic Shot Noise

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...
2. ### A Calculate variance on the ratio of 2 angular power spectra

In the context of Survey of Dark energy stage IV, I need to evaluate the error on a new observable called "O" which is equal to : \begin{equation} O=\left(\frac{C_{\ell, \mathrm{gal}, \mathrm{sp}}^{\prime}}{C_{\ell, \mathrm{gal}, \mathrm{ph}}^{\prime}}\right)=\left(\frac{b_{s p}}{b_{p...
3. ### A Computing a variance in astrophysics context

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##...
4. ### A Distance between two uncertain points using haversine?

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...
5. ### A A trend formula for the variance

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).
6. ### I Differences between the PCA function and Karhunen-Loève expansion

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...
7. ### Coin toss- expected value.

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...
8. ### Showing Rejection Region Equality with Fisher Distribution

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...
9. ### B Random Variable - Mean and Variance

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...
10. ### Prove that ##\psi## is a solution to Schrödinger equation

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...
11. ### Expected bounds of a continuous bi-variate distribution

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...
12. ### A Planck 2015 CMB temperature individual pixel variance

Hi, I'm hoping someone can point me in the right direction please. I'm using the Planck 2015 CMB temperature (intensity) SMICA pipeline maps (Nside = 2048) and am trying to determine the temperature variance of each individual pixel. Variance and hit-count were provided with the 2013 CMB maps...
13. ### Mean and Variance of a data set

Homework Statement In this problem we will be generating and analyzing lists of normally distributed random numbers. The distribution we are sampling has true mean 0 and standard deviation 1. If we sample this distribution N=5 times, what do we expect the mean to be? How about the standard...
14. ### A Stats: would the sum of the variances be 1 in this case?

Often in empirical studies you see statements that factor X explains some fraction of the variance in some other variable V, and thinking about what this means intuitively made me curious about the following question. Suppose you have a model where the values of some set of factors X1, X2, ...
15. ### Tannor Quantum Mechanics derivative of variance of position

0http://stackoverflow.com/questions/34833391/tannor-quantum-mechanics-derivative-of-variance-of-position# [Broken] In the Tannor textbook Introduction to Quantum Mechanics, there is a second derivative of chi on p37. It looks like this: χ"(t) = d/dt ( (1/m) * (<qp + pq> - 2<p><q> )...
16. ### Weighted Mean: different sample size and variance

If I have three sets of numbers A is numbers between 0 and 0.09 B is numbers between 0.091 and 0.011 C is numbers between 0.011 and 0.1 where the number of elements in A are say, 37, B are 16 and C are 178. So the three arrays have different numbers of points and different distances, plotting...
17. ### Calculate the variance

Homework Statement Find the variance of Y=3x^2+3x+3 The Attempt at a Solution Let Y = 3x^2 +3x +3 Var(Y) = Var(3x^2 +3x +3) = 9Var(x^2) +9Var(x) = 9 [E[X^4] - E[X^2]^2 +E[X^2] - E[X]^2] This is wrong.
18. ### Expected Value and Variance for Wilcoxon Signed-Rank Test

Using a normal approximation method for the Wilcoxon Signed-Rank Test, I've seen that the expected value is \mu = \frac {n(n+1)}2 and the variance is \sigma^2 = \frac {n(n+1)(2n+1)}{24} . I'm wondering why these are the expected value and variance. I do recognize the formula for the sum of...