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Questions tagged [variance]

The expected squared deviation of a random variable from its mean; or, the average squared deviation of data about their mean.

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Calculate spread (or variance) of 2D data (x,y coordinates)

I have two 256x256 images (2D data): I need to calculate the some differences between these two images. I want to calculate the spread/variance (/entropy?) of both images to compare them. How can I ...
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Can the variance of a U-statistic be of the order $O(\frac{1}{n^2})$?

It is not that easy to find estimators $T_n$ such that $\mbox{Var}[T_n] \sim O(n^{-B})$ with $B = 2$. In most cases, $B=1$.Here $n$ is the sample size. It seems, according to this paper on U-...
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How to show the inter-arrival time variance of a Cox process driven by a stationary Poisson process of constant intensity $\lambda$ is $3\lambda$

Ideas on how to show that the variance of a doubly-stochastic Poisson process(aka a Cox process) driven by a homogeneous(stationary) Poisson process of intensity $\lambda$ is $3\lambda$ ? I've come ...
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Minimum variance of the mean for $n$ correlated random variables

If $X_1,\cdots,X_n$ all have the same variance equal to 1, then $0\leq \mbox{Var}[\bar{X}]\leq 1$ where $\bar{X}=(X_1 + \cdots + X_n)/n$. The upper bound is attained if $\mbox{Cov}[X_k,X_l]=1$ for all ...
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Independent Sample Power Analysis With Unequal Means

I am running a test with two groups that have equal sample sizes (33 per group) but very unequal means - control group: 28,000 vs test group: 80,000. I am trying to determine the required sample ...
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Representing a dataset with non-normal errors

I have looked at several sources and cannot find any guidance, but perhaps I'm using the wrong terminology. I want to represent a dataset by regression line and variation similar to the way you would ...
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Are two population variances proportional if one population is derived from overlapping subsets of the other population? [on hold]

I want to rank a population of roadway segments with crash observations (rare, random events; Poisson distribution) by comparing their score using two different methods. The second method's population ...
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Mean and variance of $\tan(\mathcal{N}(\mu,\,\sigma^{2}))$

How could we find the mean and variance of $\tan(\theta )$ if $\theta \sim \mathcal{N}(\mu,\,\sigma^{2})$?
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Gradient clipping just before averaging

A typical way of implementing mini batch learning is by calculating the gradients of every element within the mini batch and then average all of these element's gradients to come up with the final ...
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38 views

Variance and covariance inequality

Given a real-valued random variable $X$, is $$2\mathbb E[X] \mathrm{Var}(X) \geq \mathrm{Cov}(X, X^2)$$ true? Any pointers for how to tackle this problem would be immensely helpful.
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Variance of the time series ARMA (1,1) model?

The ARMA (1,1) model is The variance of the white noise series is 0.09. How do you calculate Var(rt)?
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Estimate mean of a population with multiple samples when the individual sample mean is biased

I am working with datasets of grades going ~15 years back for different classes. I am trying to determine if there is a difference in the average grade for odd years compared to even years. There is a ...
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55 views

Computing Variances by Conditioning

I have trouble with the first part of this problem (Please take a look at the image below). This is an example problem from my old textbook years ago and I have had trouble understanding: How Y is ...
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What do we mean by saying “Explained Variance”

I'm studying linear regression and there is a concept I can't wrap my head around. I've heard many times the expression "the independent variable explains $a$% of the variance of the dependent ...
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Distribution of $n^{1/2}\{\hat T_n−T_n(F)\}$ in bootstrap problems

I've read this in a paper, and I don't know how to proof the last statement: Let $X_1,...,X_n$ be independent identically distributed random variables with unkown distribution function F. Suppose the ...
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Variance of ratio of a time series of random numbers

Say we have a time series($X_0, X_1, ..., X_n$) with only the first element $X_0$ known, rest being random. How can we express: $$ {\rm var} \frac{X_n}{X_0} = f({\rm var} \frac{X_i}{X_{i-1}})$$ Don'...
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SD of a likelihood function: can it replace the Standard error of a sampling distribution

I was wondering if "standard deviation" of a "likelihood function" could ever represent the "Standard error" of a "sampling distribution"? I ask this, because when one follows a Bayesian approach ...
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Interpretation of $det(X'X)$ in MLR

I would like to understand the interpretation of $det(X'X)$ in case of multiple regressors. $Var(x) = \sum_i^n(x_i-\bar{x})^2 = \frac{1}{n}\sum_i^nx_i^2 - \bar{x}^2 = \frac{1}{n}\sum_i^nx_i^2 - \frac{...
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Variance of a hypergeometric distribution

I'm trying to answer the following question from Ross's book: A pond contains 100 fish, of which 30 are carp. If 20 fish are caught, what are the mean and variance of the number of carp among the 20? ...
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Convergence of covariance matrix

I was looking for a simple way to find the number of samples $n$ needed to get a decent approximation to the covariance matrix $\boldsymbol{\Sigma}$. Given a random sample $\{ \mathbf{X}_1,\mathbf{X}...
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Unbiased variance question [closed]

A researcher is testing if a new swimming technique is more effective. She knows the average 50m time of swimmers in her club using the old technique is 35 seconds. After training 12 swimmers with the ...
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Average of several standard deviations

My aim is to calculate the measurement precision of a measuring device. To do so, I measure a part three times and calculate the standard deviation. Unfortunately, it is not possible to measure the ...
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Is there any method except Box-Cox transformations? [duplicate]

I always see that in order to reduce heteroscedasticity we can employ Box-cox transformations. But this totally useful if variance is a function of mean like $u_t^5$ or $u_t^2$....What should we do ...
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Robust estimator for process dispersion

i am currently searching for latest research on new robust estimator for process dispersion(standard deviation) to deal with outliers. we currently have Sn, MAD, Tn, Gini mean fifference, IQR.... can ...
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Finding inverse of $X'X$ in the case of two regressors [duplicate]

Variance of OLS etimator in matrix form look like this: $Var(\hat{\beta_j})=\sigma^2(X'X^{-1})$ I'm struggling to derive inverse matrix for the case with two independent variables. $X'X$ $=$ $\...
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How are variance and bias interpreted in relation to data sets

To interpret the bias we just need the training data and the test data, since it is the measure of how far off the predicted values are from the true values(test data). But, to understand the variance ...
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Bias and variance decomposition to estimate prediction

There are various ways that statisticians have come up for bias vs. variance decomposition in terms of prediction estimation. My question is this, how can one leverage or create a loss function based ...
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35 views

Degrees of Freedom In Sample Variance

Recall the formula for sample variance $$s_{n - 1}^2 = \dfrac{1}{n -1} \sum_{i = 1}^n (\bar{x} - x_i)^2,$$ where $\bar{x}$ is the sample mean. There are many proofs for why $s_{n - 1}^2$ is an ...
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How to calculate the percentage of values that lie within some range of Standard Deviation around the Mean in a Normal Distribution Curve) [duplicate]

This wiki article on 68–95–99.7 rule states - In a Bell Curve (i.e Normal Distribution Curve), 68.27%, 95.45% and 99.73% of the values lie within one, two and three standard deviations of the mean, ...
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How safe is that to ignore homogeneity of variances test and continue with Post hoc test?

I want to know if average time people spending on their favourite social media are statistically different. I consider here five groups, and the total number of participant is 700 persons. I have ...
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Difference between instantaneous and long term variance

I am studying the Heston Model which is a Stochastic Volatility Model that calculates the price of an option: $$ dS_t = \mu S_t dt + \sqrt{v_t}S_tdB_t^S$$ $$ dv_t = \kappa(\theta - v_t)dt + \xi \...
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Can the sum of the variation explained by PCA ever not match the total variance of the original data matrix?

For example if using skikit PCA when might the following not hold assuming we have a data matrix with 4 columns (features): ...
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Heteroskedasticity in linear probability models

I have a class on linear probability models. We want to estimate a model $y=\beta x$ where both $y$ and $x$ can be either $0$ or $1$, so that the conditional expectation function can be expressed in ...
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279 views

Sum of two exponential series with equal means and variances

Assuming $A$ and $B$ are two non-negative real-valued random variables such that $\mathrm{E}(A)=\mathrm{E}(B)$ (equal means) $\mathrm{Var}(A)=\mathrm{Var}(B)<\epsilon$ (equal small variances) is ...
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How does the TraMiner Package Calculate Standard Error Using Weighted Data?

The TraMiner Package includes an option to include sampling weights in the analysis. However, I haven't found any discussion in the package documentation (or associated user manual) of how standard ...
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1answer
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Why can we expand terms with random variables in the variance formula?

$\newcommand{\E}{\mathrm{E}}$$\newcommand{\Var}{\mathrm{Var}}$In the proof for showing the alternative formula for variance, i.e. $$\E[(X - \mu)^2] = E[X^2] - E[X]^2$$ I typically see the following ...
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23 views

Modeling interaction in variance

I run an experiment on both men and women participants, where I manipulate an independent variable. I know that the manipulation causes greater variance in the outcomes of the experimental group (a ...
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variance of Cohen's d effect size for two “dependent” samples [duplicate]

The derivation of the sampling variance of Cohen's $d$ effect size for the case of two independent samples is well established (see HERE). However, my question is what is the sampling variance of ...
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157 views

Calculating a sample size based on the target width of a confidence interval with stratification

I am reviewing a sampling design devised by a colleague and completely fail to understand it, although I am not a novice in statistics (but not a huge expert either). The said colleague is no longer ...
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Why does the inverse gamma distribution look essentially the same when increasing the variance?

I have a function in R which for a given mean $\mu$ and variance $\sigma^2$, spits out the parameters for the shape, $\alpha$, and rate, $\beta$ of the inverse gamma distribution with that mean and ...
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1answer
38 views

How do we deduce this fisher information relation?

Given a RS $X_{1},X_{2},\ldots,X_{n}$ whose distribution is well known (unless its parameters), how do we prove the following Fischer Information relationship \begin{align*} I_{F}(\theta) =\textbf{E}\...
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After log transforming my data to have equal variances for ANOVA, do I use the new p-values of the transformed data?

When running one-way ANOVA I did not have equal variances, so I log transformed the data and reran ANOVA with equal variances. Do I use these new p-values from Tukey's post-test on my graph showing ...
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3answers
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Prove that $\frac{1}{n(n-1)}\sum_{i=1}^{n}(X_{i} - \overline{X})^{2}$ is an unbiased estimate of $\text{Var}(\overline{X})$

If $X_{1},X_{2},\ldots,X_{n}$ are independent random variables with common mean $\mu$ and variances $\sigma^{2}_{1},\sigma^{2}_{2},\ldots,\sigma^{2}_{n}$. Prove that \begin{align*} \frac{1}{n(n-1)}\...
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4answers
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Iterated expectations and variances examples

Suppose we generate a random variable $X$ in the following way. First we flip a fair coin. If the coin is heads, take $X$ to have a $Unif(0,1)$ distribution. If the coin is tails, take $X$ to have a $...
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OLS Explained Variance

I'm using simple OLS and I want to get how much each independent variable explains variance of dependent variable. Is it possible in context of OLS or I need to use other methods?
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1answer
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Batch differences in biochemical measurements - statistical solutions?

I'm working with inflammatory markers from human participants in a current study and have been stumped by between-batch differences. One third of the participants had their inflammatory markers ...
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G efficiency in Optimal designs

I am studying Optimal Designs and found a very interesting article by Peter Goos. In the article he provides an example in the form of an Excel document of generation of Optimal Designs for various ...
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55 views

Proof that the mean is a complete sufficient statistic and the sample variance is an ancillary statistic

I have $X_1, X_2, ..., X_n $ that are random samples from the single variate $N(\mu,\sigma^2) $. I want to prove that the mean $\bar{X}$ and the sample variance $s_x ^2 = \frac{1}{(n- 1)} \sum_{i=1}^...
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1answer
29 views

Fitting a logarithmic trendline on already logged values

This is the situation. I am running trials with a population simulator, which produces various outputs (y), with the variance of these outputs being dependent on the number of clones (x) (recursions) ...
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How can I prove the following relation between the probabiloity of X and its expectation using Cauchy-Schwarz inequality?

For a random variable $ X \geq 0 $ and $E[X^2] < \infty $, I'm asked to prove the following: $ P(X> 0) \geq \frac{(E[X])^2}{E[X^2]}$ It makes intuitive sense to me that it must be the case, ...