# Questions tagged [chi-squared]

Use this tag for questions about (1) distributions of a sum of squares of independent standard normal random variables or (2) statistical hypothesis tests with such a sampling distribution if the null hypothesis is true.

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### Distributing the error in a frequency table so that the $\chi^2$ statistics are distributed according to the actual distribution?

When using the $\chi^2$ statistic, if the errors (difference between observed and expected) are too low, the resulting statistic will be low. If we repeat the experiment several times with similar ...
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### When to use chi square law for confidence intervals with mahalanobis distance?

So right now i'm reading this paper: Distance-based detection of out-of-distribution silent failures for Covid-19 lung lesion segmentation, available here: https://arxiv.org/abs/2208.03217 In brief, ...
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### Frobeinus norm of multiplication of two complex Gaussian distributed matrices

There are two complex Gaussian distributed matrices, $\mathbf{A}\in \mathbb{C}^{L\times M}$ and $\mathbf{B}\in \mathbb{C}^{N\times M}$. The elements of $\mathbf{A}$ and $\mathbf{B}$ are followed i.i.d....
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### Approximation to the CDF of chi-squared random variable

I am trying to simulate the approximation of the CDF of a chi-squared random variable, using the method proposed By Luisa Canal2006. In Luisa Canal2006, the CDF of the $\chi_n^2$, denoted as $F_n(x,n)$...
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### Can we find the closed-form solution to this optimization problem?

I have a simple optimization problem: $$\max\limits_{k}\overline{Q}_{k}^{-1}(10^{-3})-k^2\\ s.t. k>0$$ where $\overline{Q}_{k}(b)$ is the complementary CDF of the chi-...
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### Prove that XY is a chi-squared random variable if X and Y are independent normal random variables.

According to this answer, $X+Y$ and $X−Y$ are Gaussian random variables, so that $(X+Y)^2$ and $(X−Y)^2$ are Chi-square distributed with 1 degree of freedom, where $X\sim N(a,b)$ and $Y\sim N(c,d)$ ...
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### How to prove that the characteristic function of $n \bar{X}^2$ converges to that of chi square

Let $X_i$ be iid random variables with $E[X_1] = 0$ and $E[X_1^2] = \sigma^2$. I wonder how to show that the characteristic function of $n \bar{X}^2$ converges to that of $\sigma^2 Y$, where $Y$ is a ...
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### $\chi$ Distribution for Rectified Gaussians

We know that a Chi distribution is the distribution of the root of the sum of $k$ squared independent Gaussian random variables $Z_i \sim N(0, 1)$: $$Y = \sqrt{\sum_{i=0}^k Z_i^2}$$. If we have ...
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### $\chi^2$ Distribution for Rectified Gaussians

We know that a Chi-squared distribution is the distribution of the sum of squared independent Gaussian random variables $Z_i \sim N(0, 1)$: $$Y = \sum_{i=0}^k Z_i^2$$. If we have another random ...
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### $\chi^2$ test when both data sets are observed

Normally, a Pearson $\chi^2$ test can be used to help determine if an observed set of results came from a particular discrete distribution by comparing the expected and observed quantities of each ...
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### What is the set 2 ℕₒ + 1?

I encountered this notation while reading the following paper: On the efficient calculation of a linear combination of chi-square random variables with an application in counting string vacua. In it, ...
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### Why is the pdf doubled in derivation of the pdf of chi-squared distribution for 2 DOF?

I am studying this proof and I don't get why the pdf is doubled. According to Wikipedia, the pdf of A and B is expressed as below. \begin{align} f_{A,B}(a,b) &= 2f_{X,Y}(x,y) J((a,b)\rightarrow(x,...
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### What is the Integral of $\ x^a e^{ax}$?

Suppose we have the Chi-Square Probability Distribution Function (https://en.wikipedia.org/wiki/Chi-squared_test): $$f(x;k) = \frac{1}{2^{k/2}\Gamma(k/2)} x^{k/2-1} e^{-x/2}$$ I am interested in ...
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