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.
428
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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{...
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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
...
- 764
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27
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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)$...
- 135
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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-...
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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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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 $...
- 605
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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 ...
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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 $...
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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} $$.
If we have ...
- 89
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$\chi^2$ Distribution for Rectified Gaussians
We know that a Chi-squared distribution is the distribution of the sum of squared independent Gaussian random variables $Z_i \sim N(0, 1)$:
$$ Y = \sum_{i=0}^k Z_i^2 $$.
If we have another random ...
- 89
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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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Are These Two Chi-Squared Test Expressions The Same?
I am currently reading an engineering book that discusses measurements, and there is a section about the chi-squared test. As I am not an expert in statistics, I am struggling to understand why the ...
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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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1
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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 will the distribution of $u^HHH^Hu$ when elements of H follows the complex Gaussian independently and u is the eigenvector such that $\|u\|^2=1$?
I tried to solve this problem by directly expanding it. But I need some opinions on my solution. Here is the problem definition:
$\lambda=\frac{1}{|| \mathbf{H}^H\mathbf{u}||^2}\mathbf{u}^H\mathbf{H}\...
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2
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117
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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}$$
I am interested in ...
- 2,358
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Sum of squared differences between standard normal random variables
I'm struggling with the following problem:
Let $Z_1, Z_2, Z_3$ be independent, identically distributed standard normal random variables. Show that
$$\frac{1}{3}((Z_1 - Z_2)^2 + (Z_2 - Z_3)^2 + (Z_3 - ...
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How do I interpret a low Jaccard similarity and a statistically significant Chi-Squared Value?
There are two genes I am interested in. They both have many samples in which they have no expression. So I decided to see if there is a relationship between when they are expressed. I was looking for ...
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1
answer
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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 ...
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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 = ...
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0
answers
20
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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:
$x=[0.14741,0.180288,0.195,0.245342,0.25614,0.289377,0.315789,0.357143,0.431034,1.785714,2,2.323529,2....
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Reduced Chi2 for model goodness of fit
I'm currently trying to understand how I could potentially use reduced chi square ($_r\chi^2$) as a goodness of fit measure for the following (simplified) scenario:
Scatterplot of data
Whereby I have ...
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0
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49
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An Algebraic Understanding of Degrees of Freedom
I am looking at Simple Linear Regression where our model has two parameters, i.e.
$$ E[Y|x_{i}] = b_{0} + b_{1}x_{i} $$
I am having trouble understanding where the idea that $$ \frac{SSR}{\sigma^{2}}\...
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1
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41
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Calculating the Variance of the Product of Two Standard Normally Distributed Vectors
I am trying to compute the variance of the inner product of two vectors, $V_1$ and $V_2$, both of size $m$ and with independent elements that follow a standard normal distribution [$v_i \sim \mathcal ...
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Gaussian/Chi-square expectation for a rational function with high-degree term
Suppose $X \sim N(0,1)$ is a standard Gaussian random variable. How to calculate the following expectation for some $a>0$ and integer $k\ge 1$?
$$
\mathbb{E}_X\left[\frac{1}{a + X^{2k}}\right]
$$
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Can I use the Z-Test to calculate Goodness of Fit?
I have a background population defined by a normal distribution:
$$\mathcal{N}(\mu, \sigma^2)$$
where $\mu$ and $\sigma^2$ are known.
I'd like to perform a goodness of fit test on an observation to ...
- 647
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Modeling angles and magnitudes using a bi-variate gaussian.
I have a bunch of points in n-d space who's coordinates follow a Normal distribution $(X=x_1,x_2,...,x_n\sim N(0,1) )$.
The coordinates form an angle $\theta$ (with respect to some arbitrary vector $V$...
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Is the chi squared goodness of fit test valid for the multinomial distribution if the outcomes are unlabeled?
I'm investigating Sketchy Dice Inc. for purportedly selling unfair dice with the following probability distribution:
\begin{align}
p(1)&=0.1 \\
p(2)&=0.3 \\
p(3)&= 0.2 \\
p(4)&= 0.15 \\...
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Finding the non-centrality parameter of a chi-square distribution
Let $x_1, x_2, \dots, x_r$ be realization of $\mathcal{N}(\mu, \sigma)$ such that each set of size $B\leq r$ are independent of each other. Select a set of size $B$ from $x_1, x_2, \dots, x_r$. Let $...
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Chi-squared convergence to the Gaussian distribution
I'm taking a Statistics course this semester and the professor mentioned that the chi-squared distribution $\chi_n^2$ satisfies that
$$\sqrt{2\chi_n^2}-\sqrt{2n-1}$$
converges to a Gaussian ...
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1
answer
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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 ...
- 151
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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 ...
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How does the multiplication of elements in a noncentral chi-squared distribution affect the non-centrality parameter?
I have some numbers drawn from a noncentral chi-squared distribution (with a non-centrality parameter $\lambda$).
If I multiply each of my numbers by some factor F, I assume that I still have a ...
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What is the small peak I see when plotting a noncentral chi squared function?
When I try to plot a noncentral chi-squared distribution (in Python):
...
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My data doesn't fit my noncentral $\chi^2$ distribution function
I have some data that are drawn from a Gaussian distribution with mean = 0 and std = 1.
I then took each datum and squared it. The histograms below show a Gaussian distribution in black and the new ...
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Chi-squared test is failing to invalidate null hypothesis for some reason
I'm mostly a coder so apologies in advance if my math notation / explanation isn't ideal. It's been a while since college and statistics also wasn't my strong suite even back then. Would appreciate ...
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Textbooks differing on the exponent of $\sigma^2$ in $\chi^2$ distribution of $s^2$ sample variance?
These two textbooks appear to differ in their probability distribution of $s^2$ given $\sigma^2$, and I was wondering if anybody could please explain why? Specifically the exponent of ${(n-1) \over \...
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Equation for rotated elliptic Parbolloid
I am trying to do a manual Chisquare fit on a 2D data, my problem is that my data shows a rotated elliptical parabolloid. For that purpose I use the equation:
$$f(a,par)=offset + ((a - a_{0}) Cos[\...
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Proof that $VaR_c(L)=(\Phi^{-1}(\frac{c+1}2))^2$
The loss $L$ has the $\lambda_1^2$ distribution, i.e. the distribution
of the random variable $X^2$, where $X$ has a standard normal
distribution. Proof that $VaR_c(L)=(\Phi^{-1}(\frac{c+1}2))^2$, ...
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Definition of $\chi^2$-divergence between probability distributions
$
% Some definitions
\def\D#1#2{\operatorname{D_{\chi^2}}(#1 \| #2)}
\def\Df#1#2{\operatorname{D}_{f}(#1 \| #2)}
\def\E#1#2{\operatorname*{\mathbb{E}_{#1}}\left[#2\right]}
\def\dee{\mathop{\mathrm{d}\!...
- 314
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Limit of expected value of reciprocal of $\chi^2_k$ for $k \to \infty$
I must figure out $\lim_{k \to \infty} \mathbb{E}\left[ W_k \right]$ for
$$ W_k = \frac{1}{a+b \cdot\chi^2_k} \qquad a,b \in \mathbb{R}_+$$
where $\chi^2_k $ indicates a chi-squared random ...
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1
answer
150
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Distribution of sum of squared differences between two standard normal variables
The following question should be answered:
Is the distribution of $\, T_n = (X_1 - X_2)^2 + ... + (X_{n-1}-X_n)^2$, (with $X_i \sim N(0,1) \,i.i.d.)$, $\chi^2$-distributed?
For clarity, I made the ...
- 41
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89
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Chi-square issue
Chi-sq is supposed to be calculated $\sum_{i=1}^R \sum_{j=1}^C \frac{(O_{i,j} - e_{i,j})^2}{e_{i,j}}$. I thought there are two groups and it's even consistent with their chi-square estimate of which ...
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Let X(X bar) denote the mean of a random sample of size 36 for Chi-squared distribution with 2 degrees of freedom.Find P(1.75 <X<2.5)
My strategy was to prove that the given chi squared distribution converges to a normal distribution and then approximate the probability as a Normal distribution, not sure if that's the correct way to ...
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0
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34
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How to detect outliers with Gaussian mixture models?
Assuming a linear time-invariant system (LTI):
$$x_t=Ax_{t-1}+Bu_t$$
$$y_t=Cx_t+w_t$$
where $x_t$ is the system output at timestamp $t$, $u_t$ is the input to the system at timestamp $t$, $y_t$ is the ...
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0
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20
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How to obtain critical value from Chi-square pdf
I'm trying to learn about the Chi-square goodness of fit test for an assignment I'm doing, so far I've understood everything but critical values, I want to know where they come from.
Searching online ...
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53
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Expectation of a single squared term from sum of squared terms
Let $A \in \mathbb{R}^{N \times n}$ be a full row rank matrix and $b \in \mathbb{R}^{N}$ where all entries are drawn randomly, identically, and independently from $\mathcal{N}(0, 1)$. Let $f$ be the ...
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Why is Hotelling's $T^2 \sim \chi^2_p$ for large $n$?
I'm interested in some proof (simple if possible) as to why Hotelling's $T^2$ is chi-squared distributed for large n. I understand and can show that the Mahalanobis Distance is in fact chi-squared ...
