Questions tagged [chi-squared-distribution]
The distribution of sum-of-squares of k independent standard normal random variables. For the test, use the [chi-squared-test] tag. Use also for related distributions.
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How to approximate non-central chisquare distribution to Poisson weighted sum of central chi-square distribution in case of non-unit variances?
According to Statistics libre texts Equation 5.9.20, a non-central chi square distribution can be approximated as sum of Poisson weighted central chi square distributions.
$\tag{1}g(y) = \sum_{k=0}^\...
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Feature maps of the chi-squared kernel
The additive chi-squared kernel for histograms is defined as
$$K(x,y)= \sum_{i=1}^n \frac{2x_i y_i}{x_i + y_i}$$ Is this kernel positive definite on histograms? And if so, is there a known expression ...
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Confidence interval for standard deviation ($\sigma$)
I‘m trying to find 95% confidence interval for $\sigma$ from a given sample. The sample is: $$ n=6 \\40000, 200663, 142690, 48560, 40000, 242628 $$ (we know that we’re dealing with normal distribution)...
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Distribution of standard normal and squared of standard normal random variable [duplicate]
In my research I am currently conducting a monte carlo simulation, in which I end up computing the sum of say $X$ and $Y$, where $X \sim N(0,1)$ and $Y = X^2$. That is, I sum up a standard normal ...
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What is the distribution of sum of complex Normal R.Vs which are independent?
This may be a trivial question. But I am fairly new to statistics and distributions. I am trying to find analytical solution to sum of complex independendent Gaussian R.Vs following Rician ...
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Sample mean of the squares of Standard Normal R.V.s
Given $X_1,...,X_n$ are i.i.d standard normal variables, the sample mean of the squares is
$$\bar{X^2_n}=\frac{1}{n} \sum_{i=1}^n X^2_i = \frac{1}{n}(X^2_1+X^2_2+...+X^2_n)$$
For very large $n$, is ...
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Distribution of $\mathbf{x}^\top \mathbb{A} \mathbf{x} + \mathbf{b}^\top \mathbf{x}$, when $\mathbf{x}\sim\mathcal{N}$?
Let $\mathbf{x} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma)$ be a random vector following a multivariate normal density distribution. I am interested in the density function of the transformed variable
...
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Comparing two chi-square distributions
I am looking at a paper that is interested in finding a probability that one chi-squared variable is smaller than another.
More precisely
$$
\Pr\left[\sum_{i=1}^k\left(X_i-Y_i+Z_i\right)^2\leq \sum_{i=...
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How to find the expected value and median of a chi square distribution with $12$ dof in R?
How can I find the expected value and median of $X\sim \chi^2$ with degrees of freedom of $12\,$? The information I get from every other source, I find, is confusing. I am using R.
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Variance of the product of two normal variables [duplicate]
out of curiosity, I wonder if there is a solution for the expectation of the product of two chi-square variables (or the variance of the product of the normals).
Say: Let $(X,Y)$ be jointly normal, ...
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Variance of normal / chi-squared ratio
Let $X\sim N\left(0,\sigma^2\right)$ a normal variable. I am interested in the variance of:
$$Y=\frac{aX}{b+cX^2},$$
where $a,b,c>0$.
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Chi-squareness of the interaction term of the sum of square [duplicate]
I'm learning 2 way layout ANOVA.
$$
X_{ijk} = \mu + \alpha_i + \beta_j + \gamma_{ij} + u\\
u \sim \mathcal{N}(0,\sigma^2)
$$
The total sum of squares can be decomposed as followings
($*$-index ...
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Chi-Square Test vs Distribution confusion
Disclaimer: I'm new to stats so please bear with me
Everywhere I look, the Chi-Square Distribution is explained with a (Z-Score)^2 i.e
$$ ((X-\mu )/\sigma )^2 $$ and based on a random variable from a ...
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How does Pearson's cumulative test statistic approach Chi-squared distribution?
From Wikipedia,
$ \sum_1^k{Z_i^2}$ is Chi-squared distributed($Z_i$ is a standard normal random variable)
Also, it is followed by that Pearson's cumulative test statistic $ \sum_1^n{(O_i-E_i)^2 \over ...
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Independence of $\chi^2$ distribution [closed]
Suppose we are given that $X_1$ and $X_2$ are two non-negative random variable such that $$X_1+X_2\sim\chi^2_{(2)}$$. Also we are given that $X_1\sim\chi^2_{(1)}$. Can we say the following
$X_2$ is ...
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Scalar product of Gaussian random vector with projection matrix is chi-squared
We define the $n$ chi-square random variable this way : if $Z \sim N(0,I_n)$ is multivariate Gaussian random vector, then $\lVert Z \rVert ^2 = \sum_{i=1}^n Z_i^2$ (sum of $n$ standard gaussian RV ...
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Degrees of freedom of Sample Variance of Residuals (Chi-Square distribution?
In the context of jointly testing J linear restrictions, I am reviewing the distribution of the F-statistic, which is F(J, n-k). Below, R is a full row rank Jxk matrix, q is a Jx1 vector, and the null ...
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Distribution of exponent of multivariate normal distribution
My slides say that the exponent of a multivariate normal distribution, $(\mathbf{X} - \boldsymbol{\mu})^\text{T} \boldsymbol{\Sigma}^{-1} (\mathbf{X} - \boldsymbol{\mu})$, follows a chi squared ...
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What are the relevant null distributions to infer the results of a simulation that renders a multinomial distribution with three possible results?
I have simulated 10 000 occurrences of a soccer match, by generating random numbers from 0 to 1, and then passing these, and the adjusted expected goals by team for the match, through an inverse ...
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Detecting multivariate outliers with Minimum covariance discriminant and mahalanobis distance
I've read in some papers (such as this) and CrossValidated questions (such as this, that people are using mahalanobis distance based on robust estimations of location and scatter using minimum ...
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How to prove $(\hat{X}-\mu)/(\hat{S}/\sqrt{n})$ is student t with $n-1$ degrees of freedom if $X_i$ are iid $N(\mu, \sigma)$?
It is commonly stated that if $X_i$ are iid $N(\mu, \sigma)$, then with $\hat{X}$ the sample mean, and $\hat{S}$ the sample error (sample standard deviation), then $\frac{ \hat{X}-\mu}{\hat{S}/\sqrt{n}...
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Degrees of freedom of a coin toss
The probability of a fair coin landing heads is $p(H)=1/2$ and since there is only one other outcome we can deduce that the probability for tails is also $p(T)=1-p(H)=1/2$. Yet if we examine a ...
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Chi-square distribution, Chi-square distance
I am trying to use chi-square distance to find the similarity between two PDFs. My data is chi-square distributed. though the chi-square distance works well but my question is can I use the jensen-...
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Is true that the sampling distribution of $\ln \left(\chi^{2}\right)$ converges to normality much faster than the sampling distribution of $\chi^{2}$?
If true is the consequence true that $X \sim \chi^{2}(k)$ then $\sqrt{2 X}$ is approximately normally distributed with mean $\sqrt{2 k-1}$ and unit variance?
Also true that If $X \sim \chi^{2}(k)$ ...
user318514
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Is it necessary to assume normal errors and independence in a linear regression model?
I am reading Casella Berger and in the book (pg. 545), they mention that it is possible to construct a linear regression model by only assuming that:
$$
E(Y_i) = \alpha+\beta X_i \qquad \text{with} \...
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Analytical Expression for Moment of Generalized chi-squared distribution
Consider $Z\sim N(0,1)$ and the moments:
$$
E\left[ \left(C_1(Z+\sqrt{\lambda})^2 - C_2\right)^t\right].
$$
Here, $C_1$, $\lambda$ and $C_2$ are arbitrary constants which are all positive. $t$ is a ...
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CDF of product of independent chi-square distributions
Assume that we have $i=1,\ldots,n$ independent chi-square distributed random variables,
$$
X_i \sim \chi^2(k_i).
$$
What is
$$
P\left(\prod_i X_i \leq c\right)
$$
for some $c>0$?
Note: I am hoping ...
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What is the distribution of the Frobenius distance between two covariance matrices?
I am computing the Frobenius norm of the difference between two covariance matrices,
\begin{align}
|\mathbf{C}-\mathbf{C}'|_F=\sqrt{\sum_{i,j}\left(c_{ij}-c'_{ij}\right)^2}.
\end{align}
Each of these ...
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The convergence of random variables to standard normal distribution
Let $V_s$ be $n\times s$ real matrix and consisting i.i.d $\mathcal{N}(0,1)$ random variables [*]. Suppose that $O_s^1$ is the orthogonal matrix, its first column being the normalization of the first ...
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Distribution of the ratio of dependent non-central chi-square random variables
I am working on a problem that is similar to the one discussed in this link. But in my case $X_i \sim \mathcal{N}(1, \sigma^2)$, i.e., $X_i$ is not a zero-mean Gaussian RV. Specifically, I want to ...
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Finding the chi-squared distribution of a squared difference of two independent normal variables
Given two independent random variables $X\sim N(0, \sigma^2), Y\sim N(0, \sigma^2)$, what is the distribution of a variable $Q = (X-Y)^2/4$ ? What would be the expected value and variance of $Q$?
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Distribution of sample variance of independent but not identically distributed normals
When I am reading the Wikipedia page on the chi-squared distribution, it states that if $X_1, \ldots, X_n$ are $\text{N}(\mu, \sigma^2)$, then $\sum^n_{i=1} (X_i - \bar{X})^2 \sim \sigma^2 \chi^2_{n-1}...
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Overlapping $\chi^2$ random variables
I have 3 independent random variables that follow $\chi^2$ laws, with $m$ and $n$ the degrees of freedom:
\begin{align}A&\sim\chi^2_m\\B&\sim\chi^2_n\\C&\sim\chi^2_m\end{align}
I am ...
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Arithmetic with normally-distributed variables
Suppose $x \sim \mathcal N(\mu_x,\sigma_x^2)$ and $y \sim \mathcal N(\mu_y,\sigma_y^2)$ are random variables, and suppose $\mu_y$ is large compared to $\sigma_y$. I want to know about
$$
z=\frac{x}{y^...
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Why does the fact that I estimated my parameters change the way I should use chi−sqaured test?
Suppose I have a data sample and I want to test whether normal distribution with mean $0$ and variance $1$ fits this data sample. If I understand the chi-squared test correctly, I think that I should ...
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chi squared fit in 3D
I want to fit a line to $N$ 3D data points $(x,y,z)$, where $x,y$ are fitted and $z$ is fixed for all data points.
To fit the data, I use the following $\chi^2$ fit:
$$
\chi^2(a, b)_h=\sum_{i=1}^N \...
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Distribution of the approximation error in Gaussian Process Regression (finite data setting)
I am learning about Gaussian Process Regression. I would like to have some references or results regarding the distribution of the error between a given function, and the posterior obtained in ...
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Issue about both results in agreement with 2 different ways to compute variance of a random variable : weighted chisquared vs Gamma distributions
1.) I am interested in computing the variance of this observable $O$ involving the coefficients of spherical harmonics $a_{\ell m}$ and the $C_{\ell}$ which is the variance of an $a_{\ell m}$ :
$$O=\...
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How to calculate Weibull confidence interval using chi_square distribution/
I have a dataset, which I assume has a weibull distribution :
...
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Which statistical test to compare frequencies across groups?
I am running an analysis in which I want identify if there are significant differences in the features that make up two groups: people that commit crimes against children and people that commit crimes ...
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Proving $\sum_{i=1}^n(X_i-\overline X_n)^2-\sum_{i=1}^m(X_i-\overline X_m)^2 \sim \chi^2_{n-m}$
Suppose $X_1,X_2,\ldots,X_n$ are i.i.d $N(0,1)$ random variables. For $2\le m<n$, let $S_m^2=\sum_{i=1}^m(X_i-\overline X_m)^2$ and $S_n^2=\sum_{i=1}^n(X_i-\overline X_n)^2$ where $\overline X_m=\...
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Limit of $\frac{\lambda}{\chi _{2Y}^{2}}$ as $Y \sim \textrm{Poisson}(n\lambda)$ and $\lambda\to\infty.$
There are the following lines in Casella & Berger on page 438, before the equation (9.2.22):
..., write $$\lambda = \frac{\lambda}{\chi _{2Y}^{2}}\chi _{2Y}^{2}$$
where $\chi _{2Y}^{2}$ is a chi ...
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How can I find the standard deviation of the sample standard deviation from N normal distribution?
I'm an energy engineer, so my knowledge on the argument is rather limited, so forgive me in case it's a stupid question.
This question is very linked to this: How can I find the standard deviation of ...
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Understanding the relationship between the scaled inverse $\chi^2$ and inverse $\chi^2$ distributions
wikipedia says that
Also, the scaled inverse chi-squared distribution is presented as the distribution for the inverse of the mean of ν squared deviates, rather than the inverse of their sum. The two ...
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Distribution of $X'\Sigma^{-1}X$ for $X$ following a multivariate $t$ distribution
According to Golam Kibria & Joarder (2006, p.7) available here and Kotz & Nadarajah (2004, p. 19) visible in google, the distribution of $X'\Sigma^{-1}X /p$, for a known correlation matrix $\...
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Are there distributions for skewness and kurtosis? Similarly to mean (normal) and variance (chi-squared)
My question is really straightforward.
The distribution of the sample means approaches a normal distribution (CLT).
The distribution of the sample variance approaches a chi-square distribution (...
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Distribution of values of the chi square pdf
In short: if $Z$ is chi square with $n>1$ degrees of freedom and $f_Z$ its density function, and $X=f_Z(Z)$, what is the distribution of $X$ (in particular its mean and variance)?
Explanation:
I ...
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Chi-squared confidence interval for variance
When constructing, for example, a $90\%$ confidence interval for the population variance using the chi-squared distribution, we have:
\begin{align}
& P\left(a<\frac{(n-1)S^2}{\sigma^2}<b\...
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Sum of squares divided by Variance asymptotic distribution
Consider $X_i$ i.i.d. with mean 0 and variance $\sigma^2$.
I am aware that if $X_i$ is normally distributed then $\sum_{i=1}^n X_i^2/\sigma^2 \sim \chi^2_n$.
My question is, if $X_i$ has some other ...
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Distribution of the ratio of two related chi-square variates
Suppose a binormal population $X_1, X_2 \sim \mathcal{N}(0,\Sigma)$ where $\Sigma$ is obtained from $\sigma^2$ the variance of each item, assumed equal, and $\rho$, the correlation between pairs. We ...
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