**2.435 variance questions.**

Can you please help me understand how is inter-cluster (between clusters) variance defined? As opposed to intra-cluster variance which is pretty straightforward, I have not managed to found a clear ...

Here is the proof of "MVUE is unique" that my lecturer gave:
Now I understand the following:
The first expansion is done using the formula for the sum of correlated random variables (https://en....

How can one measure the accuracy of the probability distribution of, say, a physical magnitude?
I know one good candidate is the entropy, which measures the amount of information one has about the ...

Consider the following model with the usual OLS assumptions: $\epsilon_i$ are uncorrelated random variables with mean zero and constant variance $\sigma^2$.
$$y_i=\beta+ 2 \beta x_i+\epsilon_i$$
$(...

I just got some question about the hat matrix in linear models.
My first question is:
Why in a balanced one-way layout $(n_1=...=n_c=n_0)$, all leverages $h_{ii}$ have the same value $\frac{1}{n_0}$? ...

So I was trying to prove that $\operatorname{Var}[\hat{\beta}_0] = \dfrac{\sigma^2n^{-1} \sum{(x_i)^2}}{\sum{(x_i-\bar{x}})^2}$
And I got stuck with the part $\dfrac{-2\bar{x}}{\sum{(x_i-\bar{x})^2}} ...

I am trying to measure how well scores from members of a group are dispersed over possible values. Think of it as a measure of diversity function. Thus,
...

I am investigating diversity-stability relationships and
I am trying to test for the portfolio effect and insurance effect with my data.
I understand that overyielding indicates the portfolio effect ...

TL,DR: It appears that, contrary to oft-repeated advice, leave-one-out cross validation (LOO-CV) -- that is, $K$-fold CV with $K$ (the number of folds) equal to $N$ (the number of training ...

I fitted a mixed model as following:
$Y = \beta_0 + \beta_1 T + \beta_2 X + \beta_3 X T + \beta_iW_i$
$T$ is time, $X$ is my variable of interest and $W_i$ are various confounding variables. I also ...

In the classical linear regression model, the estimator of the variance of the regression error is $s^2 = \frac{e'e}{n-k} = \frac{u'Mu}{n-k}$ where u is the error vector, e is the residual vector, and ...

The title might not be perfect. But here goes:
Suppose there are 3 variables $(A,X,Y)$. And they have the following dependencies :
$Pr(Y,A,X)=Pr(Y\mid A,X)Pr(X\mid A)Pr(A)$
$A \rightarrow (X,Y)$
$...

I know how to combine two common variances with mean and sample size.
However, I don't know how to combine two variances if a variance has a special form.
From Simpson's diversity index, he proposed a ...

I'm studying clustering for system performance analysis and I'm interested in calculate the variance lost after clustering. Problem is this suppose I've 23 variables and 3000 observation. First I use ...

I am looking for a statistical method to define the variance/ diversity / inequality in a set of observations.
For example:
If I have following (n=4) observations using 4 data points, here the ...

I have a few questions regarding on how diversity is defined since I've seen differing definitions in different papers.
In the paper "Measures of Diversity in Classifier Ensembles and their ...

I have been working on the following problem:
Given you have $\text{Var}(X) = 1$, $\text{Var}(Y) = 4$, and $\text{Var}(Z) = 25$, what is the minimum possible variance for the random variable $W = X + ...

I am finding the value of a,b and c for minimising the variance of the following equation (four variables are correlated):
$$
Var(\Delta V)=V ar[\Delta S-a\Delta F_1 - b\Delta F_2 -c\Delta F_3]
$$
...

I have data about the nutrient ratio an animal wants to consume and the available nutrient ratio of grasses in it's environment. The data looks like this:
The black dots represent plant nutrient ...

There is a question on my homework that says "if variance of dependent variable changes with independent variables, what can you say about a t-test with null: b0 = 0 and alternative: b0 > 0" and I'm a ...

As per Wikipedia,
"The Net Promoter Score is obtained by asking customers a single question on a 0 to 10 rating scale, where 10 is "extremely likely" and 0 is "not at all likely": "How likely is it ...

I want to know how to calculate the variance of the function that is composed of MLE that can't estimate analytically.
Specifically, I want to know the variance of the estimate of mean of GPD that is ...

For nonrandom matrices $A(rXk)$,$ B(rXm)$, and $c(rX1)$, how does one show that
$$\newcommand{\Var}{{\rm Var}}\newcommand{\Cov}{{\rm Cov}}\newcommand{\*}{{\times}}
\Var(AX+BY+c)=A\Var(X)A′+ B \Var(...

I know that one of the standard measures for the "tailedness" of a distribution is kurtosis, i.e. fourth standardized central moment $\frac{\mu_4}{\sigma^4}$. This measure is sort of intuitive to me: ...

Unbiased weighted variance was already addressed here and elsewhere but there still seems to be a surprising amount of confusion. There appears to be a consensus toward the formula presented in the ...

Suppose we have a linear regression $Y = X\beta + e$, and we have new data come in. $Y_{new} = X_{new}\beta + e_{new}$. And we know that $Y_{new}$ and $\hat{Y_{new}}$ are independent. So $Var(Y_{new} -...

I am building a predictor for $y = f(x)$ using training samples ${(x_i, y_i)}$ (assume) drawn i.i.d from some distribution $p(x,y)$, by optimising the empirical L2-loss:
$f(x) = argmin_f \; \sum_i ||...

Hi, I have a question.
The scatter plot doesn't show any type of correlation and there is an outlier.
If the outlier was to be removed, would the correlation:
Increase dramatically
Increase ...

I see both of these equations on Wikipedia and Cross Validated being referred to as "variance":
$\mathrm{Var}(x) = \mathrm{E}\left[ \left( x - \mathrm{E}[x] \right)^2 \right]$
$\frac{1}{n} \sum_i \...

I tried the example of glmmlasso, and want to find the estimation of the variance of random error (not the random effect). I can only get the random effect. So, here is my question: How to extract the ...

How can I express the following analysis of deviance in an ANOVA table , by hand not with R..

Will someone please explain what the difference is between standard error and margin of error and when to use which? I have not had luck finding a simple explanation on the differences anywhere on the ...

I'm looking for a good tutorial about bias/variance tradeoff. In particular, I'd like to find someone that explains how different algorithms in machine learning play in this tradeoff, and possibly how ...

I am using a non linear least squares method to fit an analytical function to some experimental data. I have to provide some initial guess values to the algorithm, so I am trying to figure out how to ...

I'm using inverse-variance weighting to combine two estimates of a given data point, from different sources. Calculating the M hat is going just fine.
However, what I'm running into is that the ...

My interest is to develop a relation of the correlation coefficient when the data (both the dependent and independent variables) have measurement errors.
Intro
The measured values are related to the ...

I have a 3-level repeated measures model setup observations (level-1) nested within patients (level-2) nested within providers (level-3). One of my questions is the percentage of variance explained at ...

$Y_1, ... , Y_n$ are taken from a Poisson distribution. There are two estimators, $\mu_1 = \bar{Y}$ $\mu_2 = Y_1 + Y_2 / 2$.
Show with a mathematical proof that $\mathrm{Var}(\mu_1)$ is less than $\...

Apologies if this is a simple question; I am reviewing out of Seber and Lee's book on regression and I am pretty rusty in my linear algebra
Suppose that $X_1, ..., X_n$ have a common mean $\mu$ and ...

I have a question about concepts:
Expected value (mean value) - $μ$
Variance - $σ^2$
Standard deviation - $σ$
What is the practical meaning of these common concepts of the probability theory and ...

I want to repeatedly perform a measurement until the distribution of response values stabilizes. Is there a standard metric to stistically conclude that new data does not contain new information? I ...

Suppose that $\{X_t\}$ is a weakly stationary time series with mean
$\mu = 0$ and a covariance function $\gamma(h)$, $h \geq 0$, $\mathrm{E}[X_t] = \mu = 0$ and $\gamma(h)= \operatorname{Cov}\left(...

Consider the sum squared deviation $\text{SS}$ of $N$ samples $\hat{x}_1,\ldots,\hat{x}_N$ of a poisson random variable with mean $\mu$, i.e.
$$
\text{SS} = \sum_{m=0}^{\infty} \left( \sum_{n=1}^N \...

]1
For part B, I previously had 2.07 which is incorrect but why?! I literally followed the formula and I encountered a similar problem on the internet that did the exact same thing I did done below:
...

How to calculate the Variance and Covariance of $exp(X_{t})+2X_{t-1}$? where X is an i.i.d. normal random variables with mean zero and variance one?

A predictive model that I currently use relies on PCA with varimax rotation to reduce the dimensionality of the data (whether this is appropriate is a separate question).
The dataset consists of ...

The term error variance doesn't currently have either a tag here, or a page on Wikipedia.
p259 of Colman (2015) defines error variance as:
In statistics, the ...

What is the formula of the linearized variance of the multivariate linear regression estimator in a single-stage stratified sampling design? From the Stata manual, I know the formula for the estimator ...

The random variable $X$ takes on values -2, 0 and 2 with probabilities 1/4, 1/2 and 1/4 respectively. Find $\text{E}(X)$ and $\text{Var}(X)$.
Till this part, it was easy enough.
Then the question ...

I am asked to find mean and variance of Z. (image)
Since I am solving a preparatory examen to study, it is not clear to me how to approach the topic because I don't understand the question correctly.
...

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