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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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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Expectancy of Euclidean norm of correlated bivariate Gaussian [duplicate]

Consider a noncentral bivariate Gaussian , with nonzero mean, and with ρ not neccesarily zero, and σ1 not neccesarily equal to σ2. I want to find an expression for the expected value of the Euclidian ...
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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 $Y \sim Poisson(n\lambda)$ 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 ...
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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 ...
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7 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 ...
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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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3 answers
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How do I determine the distribution of an additional sample $x_{n+1}$ given sample mean, variance $\bar x, s^2$ of a normally distributed $x$?

Suppose I draw $n$ samples $x_1...x_n$ of a random variable $x$, which is normally distributed with an unknown mean $\mu$ and variance $\sigma^2$. From those samples, I compute a sample mean $\bar x$ ...
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What is the distribution of [constant X chi square]?

I know that (n-1)*S^2 / sigma^2 ~ chi_square(df=n-1). ※ S^2 = sigma(Xi-X_bar)/(n-1) However, if n and sigma are known, what is the distribution of S^2? As far as I think, S^2/sigma^2 = Z^2 (Z ~ N(0,1))...
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Interpretation of Chi square distribution with 1/2 degrees of freedom

How do I interpret the Chi square ditsribution with 1/2 degrees of freedom? I know that with for instance n degrees of freedom the interpretation is that this is the number of squared standard normal ...
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Equation 10.2 from The Elements of Statistical Learning. Median of a chi-squared distribution

I'm reading about AdaBoost in the The Elements of Statistical Learning and I don't understand the equation 10.2. Below is an excerpt from the book. The power of AdaBoost to dramatically increase the ...
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Show that noncentral $X_2$ is $\chi^2(r-r_1,\theta -\theta_1)$

As I read a book named 'Introduction to mathematical statistics' written by Hogg et al, I stuck with the below question. $X_1$ and $X_2$ be two independent random variables. And Let $X_1$and $Y=X_1+...
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1 vote
1 answer
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2 approaches for Monte-Carlo : weighted sum of $\chi^2$ distribution and Moschopoulos distribution with Gamma distribution

If I take as definition of $a_{lm}$ following a normal distribution with mean equal to zero and $C_\ell=\langle a_{lm}^2 \rangle=\text{Var}(a_{lm})$, and if I have a sum of $\chi^2$, can I write the 2 ...
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Variance in variance-weighted variance estimate?

Apologies for the confusing title, but I couldn't resist. Much can and has been said about computing the unbiased variance using a sample of points, weighting by the variances of each point (for ...
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For Poisson GLMs, when does the residual deviance follow a chi square distribution?

According to Generalized Linear Models by McCullagh and Nelder (I am looking at the 2nd edition, 1999), the deviance function is defined as $$D(y; \hat{\pi}) = 2[l(\tilde{\pi}; y)- l(\hat{\pi}; y)]$$ ...
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2 answers
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confused about unbiasedness of sample mean

Recently, in a different thread, I was convinced by others ( after claiming that they were wrong ) that the sample mean is unbiased for the mean of its underlying distribution. But the case of the chi-...
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Properties of $\Gamma$ distribution function that allow to include or not terms depending on multipoles - Computing variance [duplicate]

In the context of harmonic spherical decomposition where $C_\ell$ is the variance of $a_{lm}$ for a given multipole $\ell$, i.e $C_\ell = \langle |a_{lm}^2|\rangle$, I need help about the $\chi^2$ ...
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1 vote
1 answer
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Expectation and distribution of ratio of correlated Gamma/Chi-square random variables

This question is very similar to: Distribution of the ratio of dependent chi-square random variables But the big difference is what happens when we don't have standard normal variables. I want to ...
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SaddlePoint Approximation Score Statistic

My question will be very practical. Actually, I work in statistical genetics and I have 1000 Score statistics for which I want to compute the corresponding p-values. To be more specific, I suppose a ...
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Deriving the Chi Square distribution [duplicate]

Let $X_1,...,X_n$ be IID random variable such that $X_i \sim N(0,1)$ for all $i=1,...,n$. Derive the distribution (i.e. give the pdf/cdf) of $\sum_{i=1}^{n} X_i^2$. I know that from the definition of ...
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Distribution of the sample mean Poisson's occurrence rate

Simple question. Given a sample of 100 values, the $\hat\lambda$ is calculated (e.g. with poissfit in Matlab), and the confidence intervals are provided. What's the ...
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4 votes
1 answer
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Derivation of F-distribution from inverse Chi-square?

I am trying to derive F-distribution from Chi-square and inverse Chi-square. Somewhere in process I make a mistake and result slightly differs from the canonical form of Fisher-Snedecor F distribution....
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Why does one compute an outer product of marginal distributions (of contingency table) instead of splitting data up completely equally?

I am studying about the chi-squared test statistic and came across the following code. ...
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SOLVED: What is the distribution of the squared exponential of two gaussian distributions of odd dimension?

Let $X,X'$ follow the $n$ dimensional Gaussian distribution (zero mean unit variance for simplicity). My question is: what is the distribution of $Y = e^{-\|X-X'\|^2}$, particuarly when $n$ is odd? $...
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