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.
455 questions
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Chi-squared random variable
Given a random variable $X$ which is $\chi^2_{n}$, can I define $n$ independent standard normal random variables $Z_{1,...,n}$ on the probability space such that $X = Z_1^2 + Z_2^2 + ... + Z_n^2$ ...
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F-test: Is the $\chi^2$ difference of nested models $\chi^2$-distributed?
I have recently started working with F-testing for determining optimal polynomial order of a fit. I have calculated the $\chi_k^2$ for a certain model with $k$ parameters and $\chi^2_{k'}$ with $k'>...
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Identifying the distribution of a random process associated to the square root process
Let $x(t)$ be a random square-root process that follows
$$dx(t) = (a + bx(t)) \, dt + c\sqrt{x(t)} \, dW(t)$$
where $W(t)$ is a standard Brownian motion for some filtration.
This has many names ...
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Question about Unbiasedness of Raw Sample Moments, Odd Simulation Results
I'm performing a Monte Carlo simulation to confirm that $$m_k'=\frac{1}{n}\sum_{i=1}^n X_i^k$$ is an unbiased estimator for $\text{E}[X^k]$. In the event $X\overset{iid}{\sim} \chi_1^2$, we have $\...
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issues when constructing a confidence interval using chi-squared distribution
Imagine a randomized controlled trial with a binary treatment and serveral covariates $X_1,...,X_n$.
After randomization usually a balance comparison is made betwenn the two treatment group regarding ...
user392103
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Condition for quadratic form to have a chi-squared distribution
According to Seber and Lee's textbook theorem 2.8: If $\mathbf{Y} \sim N_n(\mathbf{0},\Sigma)$ and $A$ is symmetric then $\mathbf{Y}^TA\mathbf{Y} \sim \chi^2_r$ if and only if $r$ of the eigenvalues ...
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Hypothesis testing with unknown covariance matrix under the alternative hypothesis
I am trying to solve a binary hypothesis testing problem of the form:
$H_1 : \mathbf{y} = \theta \cdot \mathbf{x} + \mathbf{n}$
$H_0 : \mathbf{y} = \mathbf{n}$
where $\theta \in \mathbb{R}$ under $H_1$...
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chi square and exponential distributions
The plot of a single chi square RV approaches infinity at X^2 = 0. Why is that when chi square RVs are added, the resulting pdf plot seems to be at a finite value at Y= 0, where Y = the sum of these ...
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Degrees of freedom for Ljung-Box test
I have two questions regarding the degrees of freedom for the Ljung-Box test on residuals in case of different AR(p) models:
In case of a model with non-consecutive lags: As I understand it, one has ...
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Lower Tail Bound for Noncentral chi-squared distribution
A noncentral chi-squared random variable $Z$ is of the form
$$
Z = \sum_{i=1}^k X_i^2
$$
where $(X_1, X_2, \ldots, X_i, \ldots, X_k)$ be $k$ are independent, normally distributed with mean $\mu_i$ and ...
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Why do we usually test independence of row/column variables of a contingency table with a chisquare statistic?
For an $r \times c$ contingency table, the cells containing observed frequencies, the null hypothesis "the row and column variables are independent" is typically tested by using a $\chi^2$ ...
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Expected Value Chi Square distribution
I'm trying to simulate the distribution from the sample variance $s^2$ and compare it with the theoretical distribution.
Therefore, I perform a fairly simple simulation (upfront, I'm not a ...
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Why does $E(V_n/(n+2)-1)^2=2/(n+2)$ when $V_n\sim\chi^2(n)$?
I was reading some lecture notes when I saw a simplification I didn't understand. Specifically, we have $V_n\sim\chi^2(n)$. It was then written then
$$E\left(\frac{1}{n+2}V_n-1\right)^2=\frac{2}{n+2}.$...
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Distribution of norm of fixed vector projected onto a Gaussian subspace
Let $\Sigma \in \mathbb{R}^{m \times m}$, $\Theta_0 \in \mathbb{R}^{k \times m}$, $v = \Theta_0 \beta \in \mathbb{R}^k$ with $\| v \| = 1$ and $\Theta \sim \mathcal{N}(\Theta_0, \mathrm{Id}_k \otimes \...
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How to Find Probability Sum of Squares of Std Normal is greater than Sum of Squares of Non-Standard Normal with Mean 0
I'm looking for the probability that the sum of squares of a standard normal is less than the sum of squares of a non-standard normal with mean 0 and fixed std-dev.
Lets say $X_i \sim \mathcal{N}(0,1)$...
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