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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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 $X \sim \chi^{2}(k)$ ...
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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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2 answers
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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 $Y \sim Poisson(n\lambda)$ and $𝜆 → ∞$

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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Independence of Random Variables - from t distribution and its definition [duplicate]

I have a somewhat strange question about the independence of random variables. It comes from the definition of t-distribution. In this definition, we need two independent random variables and we can ...
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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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Expected distance under Gaussian noise

Summary I'm working on a tracking problem, where I'm trying to estimate the position of an object that moves in on plane. In my simulator, at each sampling step I generate a measurement that is given ...
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whether $T_3$ has a chisquare distribution subject to a multiplicative constant

Suppose $X_1,...,X_n$ are random samples from $N(0, \sigma^2)$, and $\bar X_n = n^{-1}\sum_{i=1}^{n}X_i$. Let $Y =$ $Y_1 \choose {Y_2}$, where $Y_1 = X_1 - \bar X_n$, $Y_2 = X_2 - \bar X_n$, be a ...
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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 ...
3 votes
1 answer
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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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Significance test of improvement of clustering methods compared to reference - correct idea?

Performing cluster analysis, I have a reference for the results and the results of two methods A and B. I am able to calculate fitness metrics (like adjusted mutual information) between the reference ...
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How to perform significance test between two contingency tables? [duplicate]

Performing cluster analysis, I have a reference for the results and the results of two methods A and B. I am able to calculate fitness metrics (like adjusted mutual information) between the reference ...
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PDF of $x_1^2+x_2^2-x_3^2-x_4^2$ with $x_i \sim N(\mu_i,1)$

What is the probability distribution function of the random variable $X$ $X=x_1^2+x_2^2-x_3^2-x_4^2, \tag{1}$ where $x_i$ are independent normally distributed variables $\mathcal{N}(\mu_i,1)$? What I ...
8 votes
1 answer
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Median of the sum vs. sum of the median for Gaussian variables

This is a problem I stumbled upon in my research. Consider $n$ Gaussian random variables $x_i \sim \mathcal{N} (\mu_i, \sigma_i^2)$, each with its own mean $\mu_i$ and variance $\sigma_i^2$. Can we ...
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Can chi-squared distribution be left-skewed?

I'm learning chi-squared distribution and know that as degree of freedom increases, right-skewed chi-squared distribution will approximate to normal distribution shape. I'm wondering can its ...
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Expectation of square root of scaled noncentral chi-squared minus a constant

Suppose that $X\sim \mathcal{N}(\mu,\sigma^2)$ and $\mu>\sqrt{c}$. Is there an approximate closed expression for $\mathbb{E}[\sqrt{X^2-c}]$? My application allows me to impose an upper bound on the ...
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Difference distribution of iid noncentral chi2 variables

Given are two identically distributed independent noncentral chi-square variables $$Z_1\sim \chi^2(k,\lambda)\\Z_2\sim\chi^2(k,\lambda)$$ with $k=2$ degrees of freedom and location parameter $\lambda$....
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Variance of a normal random variable when conditioning on a correlated normal random variable being above a threshold

Suppose $X$ and $Y$ are correlated with correlation coefficient $\rho$. They are jointly normal with means $\mu_X$ and $\mu_Y$ respectively. Then what is $Var[X | Y \geq T]$? Feel free to add ...
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Does chi-square with N degrees of freedom stochastically dominate chi-square with M degrees of freedom whenever N>M?

Does chi-square with N degrees of freedom stochastically dominate chi-square with M degrees of freedom whenever N>M? My guess is yes. As $\chi^2_{N}=\sum_{i=1}^NZ_i^2$ and $\chi^2_{M}=\sum_{i=1}^M ...
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Validity of a joint distribution and the associated Likelihoods

I want to check the validity of a code that produces a 2D contour between 2 parameters with their associated Likelihood (normalized). Here's an example below : As you can see, I have plotted, for 3 ...
3 votes
1 answer
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Distribution of $\frac{Z_1^2 + Z_2^2}{Z_1+Z_2}$ where $Z_1, Z_2$ are standard normals

Let $Z_1, Z_2 \sim \mathcal{N}(0,1)$ be i.i.d random variables. I wish to find the distribution of \begin{align} \frac{Z_1^2 + Z_2^2}{Z_1+Z_2} \,. \end{align} It is well known that $W = Z_1^2 + Z_2^2 ...
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Properties of $\chi^2(1)$ multiplied by a real value "$a$"

Is it true that the $\chi^2$ distribution with $k=1$ (noted $\chi^2(1)$) multiplied by a real value $a$ is equal to $\chi^2(a)$ ? If not, is there a particular distribution for $a\cdot\chi^2(1)$? If ...

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