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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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Cumulative probability of $X \sim \chi^2$ on the interval $[0, \mathbb{E[X]}]$

I am working on the $\chi^2$ distribution and have the following assumption: The probability of the cumulative distribution function of a $\chi^2$ distributed random variable is greater than $\frac{1}{...
wim15's user avatar
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Expectation of Inverse-chi-squared distribution with 1 degree of freedom

I need to find the expectation of $1/z^{2}$ where $z \sim \operatorname{N}\left(0, 1\right)$:...
Gilad Deutsch's user avatar
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Pearson's Chi-Square test, set distribution into categories but desired property within

I have a dataset which falls into 4 categories and I would like to use a Pearson's Chi-square test but I am unsure about the underlying parameters for the Chi-square test statistic. The scenario is as ...
raffaelluca's user avatar
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Numerical comparison of two probability distribution functions

I have a dataset which satisfies Benford's law of anomalous numbers. However, I proved numerically that each sample in the dataset satisfies Benford's law but the probability density function of the ...
Lazar Šćekić's user avatar
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Cumulative distribution function of chi-squared distribution

I am trying to understand the proof here. At some point goes from $$F_Y(y) = \frac{F_Y(y)}{\mathrm{d}y} \mathrm{d}y = f_Y(y) \mathrm{d}y \tag{1}$$ to $$f_Y(y) \mathrm{d}y = \int_{V} \prod_{i=1}^{k} \...
Thoth's user avatar
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expectation of logarithm of scaled chi-squared random variable

Suppose $x_1,\dots, x_n\sim N(0,1)$ are iid Gaussians, and $\lambda_1,\dots,\lambda_n > 0$ are positive scalars. Define scaled chi-squared random variable $Y = \sum_i^n \lambda_i x_i^2,$ where we ...
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How to determine if a race course is fair (on the open water)?

I am analyzing the distribution of race results using statistical methods like chi-square tests to look for the most uniformly distributed results. I have data from a number of different locations and ...
user6972's user avatar
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How to calculate mean and variance for square root of chi-squared distribution divided by the degrees of freedom

Distribution of Process capability (Cp) is given by the below formula. $$Cp\sqrt{\frac{\chi_\nu^2}{\nu}}$$ $\nu$ = The degrees of freedom To calculate mean and variance of $\sqrt{\frac{\chi_\nu^2}{\nu}...
Steve's user avatar
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28 views

What is the distribution of the observed counts of the Chi-squared test?

I was reading this answer, trying to get some intuition for how Pearson's chi-squared test works: https://math.stackexchange.com/a/2074074/1226290 Everything makes sense from this answer, except for ...
KDJ's user avatar
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Finding α-Quantiles of χ2 Distribution for Variance Estimation

I posted this same question yesterday but it got closed because i hadn't met the guidlines for questions, my apologies guys, so i'm going to re-write it better this time. Exercise 17. Let $X_1, \ldots,...
mathmath's user avatar
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A integer calculation problem whose integrand is $\frac1{A \chi^2+B}$

I'm calculating an expectation of the form $$\mathbb E\left[1\over aX+b\right]$$where $X\sim\chi^2_1(0)$ obeys an central chi squared distribution with 1 degree of freedom. The integral formula is $$\...
ShuchenSean's user avatar
2 votes
1 answer
45 views

the expectation of linear combination of chi-squared random variables with 1 degree of freedom

I'm calculating the problem descripted in the title, and found it a little bit hard, here is the problem: Suppose $X_i\sim\mathcal{N}(0,1)$ is standard normal random variables, now we need to ...
ShuchenSean's user avatar
1 vote
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Square of a Chi-squared Random Variable

I have a Chi-squared RV with 2NL degrees of freedom and I am interested in the distribution of its square ($Y=X^2$). I have tried the transformation method to get the expression for the pdf of Y as: $...
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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 ...
Chris Vilches's user avatar
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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, ...
levo-str's user avatar
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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....
kyub's user avatar
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2 answers
165 views

Asymptotic convergence of sampling distribution of the sample variance

Let's consider a set $\{X_i\}_{i=1}^N$ of $N$ i.i.d. random variables drawn from the distribution $P_X(x) = \mathcal{N}(\mu, \sigma^2)$. Define the variable $$\hat{\sigma}^2 = \frac{1}{N} \sum_i (X_i -...
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suitable statistical method for variables dependency testing

I made a survey among small and medium enterprises and I asked two questions: Q1: How many employees does Your enterprise have ? none, 2. 1 to 9, 3. 10 to 19, 4. 20 to 49, 5. at least 50. Q2: How ...
elliptic's user avatar
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Statistical test for null hypothesis $\|p-q\|\le \epsilon$ for $k$-dimensional categorical data

Is there any known (asymptotic) statistical test for the null hypothesis $$\|p-q\|\le \epsilon$$ for $k$-dimensional categorical data independently taken from two societies for some given norm $\|\...
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1 answer
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chi2 probability

I am trying to use this equation $Prob_d(\chi^2 > \chi_0^2) = \frac{2} {2^{d/2} \Gamma(d/2)} \int_{\chi_0}^{\inf} x^{d-1}e^{-x^2/2} dx$ to compute probabilities of an empirical distribution $\chi^...
konstanze's user avatar
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pdf of Linear Combination of the same random variable

Let's say that a random variable X has a probability p to be Gamma($\alpha,\beta$) and a 1-p probability to be $\chi^2$(r). How do I prove that $f_x(x) = p \frac{1}{\Gamma(\alpha)\beta^\alpha}x^{\...
Albert Wijaya's user avatar
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What test can we use to compare the sample proportion of multiple dependent samples (> 2) for non-dichotomous data (> 2 categories)?

I am currently studying hypothesis testing for dependent two-sample (proportion). The crux for my question is this, what test does one use to compare the proportion of multiple samples for non-...
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Transform Wishart distribution to Chi-square distribution

This's actually what I'm trying to prove: $$ \frac {a^{'}\Sigma^{-1}a}{a^{'}W^{-1}a} \sim \chi^{2}_{n-p+1} $$ $a$ is any P-dimensional nonzero constant vector, and $W \sim W_{p}(n,\Sigma)$, $\Sigma$ ...
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Given the L2 norm of a Gaussian matrix, what distribution does the Gaussian matrix follow?

Given a random gaussian matrix X with zero mean matrix and covariance matrix Σ, and two deterministic matrices A and B. If I know the value of $||{\bf{AX}}||_F^2$, how could I get the pdf of $||{\bf{...
Sheperd Lv's user avatar
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54 views

Distribution of the square of the euclidean norm of a gaussian vector with diagonal covariance matrix

Let $X\sim\mathcal{N}_d(0,\Sigma)$ be a $d$-dimensional gaussian vector, where $0\in\mathbb{R}^{d}$ and $\Sigma\in\mathbb{R}^{d\times d}$ is diagonal. I'm interested on the distribution of: $$ ||X||^2 ...
MathRevenge's user avatar
1 vote
1 answer
93 views

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 $F_n(x,n)$...
Tyke's user avatar
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Can we find the closed-form solution to this optimization problem?

I have a simple optimization problem: \begin{equation} \max\limits_{k}\overline{Q}_{k}^{-1}(10^{-3})-k^2\\ s.t. k>0 \end{equation} where $\overline{Q}_{k}(b)$ is the complementary CDF of the chi-...
Tyke's user avatar
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0 answers
150 views

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 ...
joy's user avatar
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regarding $\chi^2$ is sub-exponential

I'm currently studying the fact that $\chi^2$ is sub-exponential, which seems similar to this site, and what I'm confused looks similar to what the author of the link was confused as well. Show that $...
jason 1's user avatar
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Eigenvalues of multivariate t-wishart and wishart matrix

If $W$ is a wishart matrix with Identity Covariance and $n$ degrees of freedom. And another matrix $X= v*W*S$ where $S$ is a diagonal matrix with diagonal elements as iid inverse-$\chi^2$ distributed ...
maddy's user avatar
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Best estimate of parameter in a Chi-squared test causes us to reject $H_0,$ but another value of the parameter causes us to accept $H_0$.

For a $\chi^2$ goodness-of-fit test where a parameter is estimated (for example, $p$ is estimated if you're testing to see if the Binomial distribution is a good fit to the data), is it true that if $...
Adam Rubinson's user avatar
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1 answer
52 views

$\chi$ 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 ...
Liam F-A's user avatar
2 votes
3 answers
125 views

$\chi^2$ 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 ...
Liam F-A's user avatar
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0 answers
24 views

$\chi^2$ 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 ...
DongKy's user avatar
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1 answer
72 views

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, ...
RMS's user avatar
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1 answer
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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,...
CatsEyeblow's user avatar
3 votes
2 answers
144 views

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 ...
stats_noob's user avatar
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1 answer
70 views

Sum of squared differences between standard normal random variables

I'm struggling with the following problem: Let $Z_1, Z_2, Z_3$ be independent, identically distributed standard normal random variables. Show that $$\frac{1}{3}((Z_1 - Z_2)^2 + (Z_2 - Z_3)^2 + (Z_3 - ...
user2296226's user avatar
0 votes
1 answer
26 views

asymptotics of the percentage point for $\chi^2_k$

For $k\geq 1$ let $T_k\sim \chi^2_k$. For $a\in ]0,1[$ let us define $c_a(k) = \min \{ s\geq 0, \mathbf P ( T_k \leq s )\geq a\}$. Would anybody have a free reference for a study of the asymptotics ...
jo_le_clodo's user avatar
-2 votes
1 answer
57 views

Is there a significance test for testing that the population standard deviation is greater than zero?

I am working on a project where I need to test the following hypotheses: $H_0: σ = 0 H_a: σ > 0$ I am aware of the chi square test for standard deviation, that is $χ^2 = ...
efish1824's user avatar
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0 answers
21 views

Fitting a set of points to a distribution by adding up to three degrees of freedom with Python

I have a set of points whose shape is as below Its set of $x$ and $y$ points is as follows: $x=[0.14741,0.180288,0.195,0.245342,0.25614,0.289377,0.315789,0.357143,0.431034,1.785714,2,2.323529,2....
user3341263's user avatar
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1 answer
55 views

Calculating the Variance of the Product of Two Standard Normally Distributed Vectors

I am trying to compute the variance of the inner product of two vectors, $V_1$ and $V_2$, both of size $m$ and with independent elements that follow a standard normal distribution [$v_i \sim \mathcal ...
Peyman's user avatar
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0 votes
1 answer
31 views

Gaussian/Chi-square expectation for a rational function with high-degree term

Suppose $X \sim N(0,1)$ is a standard Gaussian random variable. How to calculate the following expectation for some $a>0$ and integer $k\ge 1$? $$ \mathbb{E}_X\left[\frac{1}{a + X^{2k}}\right] $$
Nick's user avatar
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1 answer
107 views

Can I use the Z-Test to calculate Goodness of Fit?

I have a background population defined by a normal distribution: $$\mathcal{N}(\mu, \sigma^2)$$ where $\mu$ and $\sigma^2$ are known. I'd like to perform a goodness of fit test on an observation to ...
Connor's user avatar
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1 vote
0 answers
35 views

Modeling angles and magnitudes using a bi-variate gaussian.

I have a bunch of points in n-d space who's coordinates follow a Normal distribution $(X=x_1,x_2,...,x_n\sim N(0,1) )$. The coordinates form an angle $\theta$ (with respect to some arbitrary vector $V$...
Liam F-A's user avatar
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0 answers
31 views

Is the chi squared goodness of fit test valid for the multinomial distribution if the outcomes are unlabeled?

I'm investigating Sketchy Dice Inc. for purportedly selling unfair dice with the following probability distribution: \begin{align} p(1)&=0.1 \\ p(2)&=0.3 \\ p(3)&= 0.2 \\ p(4)&= 0.15 \\...
Chessnerd321's user avatar
3 votes
2 answers
427 views

Chi-squared convergence to the Gaussian distribution

I'm taking a Statistics course this semester and the professor mentioned that the chi-squared distribution $\chi_n^2$ satisfies that $$\sqrt{2\chi_n^2}-\sqrt{2n-1}$$ converges to a Gaussian ...
Frank William Hammond's user avatar
0 votes
1 answer
358 views

How do I calculate the parameters of a non-central chi-squared distribution if I know everything about my original Gaussian distribution?

I generate some Gaussian data (10,000 points) with (for example): mean = 3.0 std = 1.0 And I fit a Gaussian function to it: So far so good. Next I square each of my 10,000 data points. I believe that ...
user1551817's user avatar
0 votes
1 answer
442 views

How to find probabilities for $\chi^2$ distribution

I apologize if I didn't structure the question good but english isn't my first language so I hope you understand but I will try my best to be very clear. I wanted to know how can I use a chi squared ...
randomvlad's user avatar

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