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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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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Distribution of the correlation coefficient based on quadratic forms
Let $x,y$ be two independent random correlated vectors following the same multivariate (real or complex) centred normal distribution, and let $A$ be a non-negative linear operator.
We can read here, ...
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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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Projection of i.i.d. Gaussians onto correlated Gaussian is bounded from below by chi-squared
Let $\varepsilon \sim \mathcal{N}(0, \mathrm{Id}_k)$ and $\varepsilon_0 \in \mathbb{R}$ with $\varepsilon_0 \neq 0$.
For some matrix or vector $A \in \mathbb{R}^{k \times p}$, let $P_A := A (A^T A)^{-...
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If $X \sim\textrm{ Bin}(100, 0.5), $ then what is the approximate distribution of $(X/5 - 10)^2? $
I am not able to solve this using transformation since binomial does not have a CDF. The question has 4 options, so I tried calculating the expectation of this and then comparing it to the ...
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Relationship Between Chi-Square/Gamma & t/lst distributions?
I'm trying to understand $\chi^2_n$ & $\Gamma(\theta, k)$ distributions. Currently I believe they're comparable to t (aka $t_v$) & location-scale-t (aka $lst(\mu, \sigma^2, v)$) distributions ...
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$\chi^2$-distribution of $(x_1-x_2+x_3+x_4)^2-(-x_1+x_2+x_3+x_4)^2$ with $x_i \sim N(\mu_i,\sigma_i)$
Given are 4 independent normal variables $x_i \sim \mathcal{N}(\mu_i,\sigma_i)_{i=1,\ldots,4}$, that are used to define the random variable
$$Z\sim (x_1 - x_2 + x_3 + x_4)^2 - (-x_1 + x_2 + x_3 + x_4)^...
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Calculating the empirical mean of given random variable using R
I was reading a Wikipedia page regarding the chi-square distribution: Chi-squared distribution (Wikipedia).
It says the expected value of a chi square RV equals k where k is the dgf. I've tried to ...
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On an infinite linear combination of chi-squared random variables
Question
Let $Z_i\sim\chi_{(1)}^2$ be i.i.d. chi-squared random variables with 1 degree of freedom. We define:
$$
W_{\infty} = \sum_{k = 1}^{\infty} \frac{Z_k}{2^{k}}
$$
I have interest in computing ...
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How to calculate confidence intervals for very small sample point estimates (percentages) for a large positive distribution? [closed]
I need help on breaking down a confidence limit formula in Excel. The file is calcuating confidence limits for very small point estimates from large samples and for large population that are ...
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Moments of sum of squares of independent gaussians $X_i \sim \mathcal{N}(\mu_i,\sigma^2_i)$, or $||X||^2$
Say that we have $X_i \sim \mathcal{N}(\mu_i, \sigma_i^2)$. Is there some formula to calculate analytically the expected value of the sum $S = \sum_i^n X_i^2$?. This is equivalent to computing $\...
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Why are the p-values for a binomial test very different from the p-values for a chi-squared test?
I am testing if a coin is fair by throwing it n-times and having n/2 + sqrt(n) heads.
However, I get very different p-values ...
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In simple linear regression model, why do we calculate the confidence interval for slope parameter using t-distribution? [duplicate]
I'm taking a regression analysis course and we were studying simple linear regression.
I've understood how slope
$$
\hat\beta_1 follows \space N(0,\sigma^2 / S_{xx})
$$
and is normally disributed.
And
...
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What kind of logrank test is performed by survdiff function in R?
Information added: this question is not duplicate, neither my question nor the question in the link had been solved at all, I added more info to see if you can see the problem
I was trying to ...
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The test point’s standard deviation from the origin (ESL’s Exercise)
This is Ex.2.4 from The Elements of Statistical Learning.
I don’t understand the sentence that I underline in the image.
I know that $\sqrt{10}$ is approximately equal to 3.1, but I don’t know how to ...
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Is it true that if $X$ follows central chi-square distribution then $X+a$ follows non-central chi-square distribution where $a$ is positive constant?
Is it true that if $X$ follows a central chi-square distribution, then $X+a$ follows a non-central chi-square distribution where $a$ is a positive constant?
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CDF for squared sum of Rayleigh random variables
In short, I am looking to estimate the distribution of
$ \eta = \sum_{i=1}^N (X_i - z_i)^2$, for each $X_i \sim \text{Rayleigh}(1)$ and constants $z_i$.
If $X_i$ were Gaussian, then this could be ...
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Compare two contingency tables with the same structure
I want to compare two contingency tables A and B with the same structure (same columns and rows). My first idea was to use chi squared distance $\sum_{i=1}^{p}{\frac{(O_i-E_i)^2}{E_i}}$. But, some ...
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Am I using the chi-squared test correctly?
I have a set of $2,142$ measurements of some value that are grouped into $18$ bins of equal length (according to the value measured). I want to check the resulting distribution for uniformity.
As far ...
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How do I determine which events occurred significantly more during a certain operating mode?
I have data of $N$ different event $E_n$ occurring at times $t_{n,i}$. So any event $E_n$ could have occurred any number of times.
Normally the system operates in operating mode A. Sometimes the ...
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Distribution for an infinite sum of weighted chi-squared distributions
Let $X_1,X_2,\ldots$ be an infinite list of independent normal variables $X_i\sim\mathcal N(0,1)$, for $i=0,1,\dots$. Consider the the sum
$$Y=\sum_ir^iX_i^2,$$
where $r\in(0,1)$ is a parameter to ...
user340474
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sum of noncentral Chi random variables
if $X_1,...,X_n$ are independent random variables with noncentral chi distributions (same $df$ but different $\lambda$),
What is the distribution of $\sum_{i=1}^{n}{X_i}$
Just wondering if it can be ...
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How to approximate non-central chisquare distribution to Poisson weighted sum of central chi-square distribution in case of non-unit variances?
According to Statistics libre texts Equation 5.9.20, a non-central chi square distribution can be approximated as sum of Poisson weighted central chi square distributions.
$\tag{1}g(y) = \sum_{k=0}^\...
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Feature maps of the chi-squared kernel
The additive chi-squared kernel for histograms is defined as
$$K(x,y)= \sum_{i=1}^n \frac{2x_i y_i}{x_i + y_i}$$ Is this kernel positive definite on histograms? And if so, is there a known expression ...
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Confidence interval for standard deviation ($\sigma$)
I‘m trying to find 95% confidence interval for $\sigma$ from a given sample. The sample is: $$ n=6 \\40000, 200663, 142690, 48560, 40000, 242628 $$ (we know that we’re dealing with normal distribution)...
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Distribution of standard normal and squared of standard normal random variable [duplicate]
In my research I am currently conducting a monte carlo simulation, in which I end up computing the sum of say $X$ and $Y$, where $X \sim N(0,1)$ and $Y = X^2$. That is, I sum up a standard normal ...
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Sample mean of the squares of Standard Normal R.V.s
Given $X_1,...,X_n$ are i.i.d standard normal variables, the sample mean of the squares is
$$\bar{X^2_n}=\frac{1}{n} \sum_{i=1}^n X^2_i = \frac{1}{n}(X^2_1+X^2_2+...+X^2_n)$$
For very large $n$, is ...
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Distribution of $\mathbf{x}^\top \mathbb{A} \mathbf{x} + \mathbf{b}^\top \mathbf{x}$, when $\mathbf{x}\sim\mathcal{N}$?
Let $\mathbf{x} \sim \mathcal{N}(\boldsymbol{\mu}, \Sigma)$ be a random vector following a multivariate normal density distribution. I am interested in the density function of the transformed variable
...
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How to find the expected value and median of a chi square distribution with $12$ dof in R?
How can I find the expected value and median of $X\sim \chi^2$ with degrees of freedom of $12\,$? The information I get from every other source, I find, is confusing. I am using R.
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Variance of the product of two normal variables [duplicate]
out of curiosity, I wonder if there is a solution for the expectation of the product of two chi-square variables (or the variance of the product of the normals).
Say: Let $(X,Y)$ be jointly normal, ...
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Chi-squareness of the interaction term of the sum of square [duplicate]
I'm learning 2 way layout ANOVA.
$$
X_{ijk} = \mu + \alpha_i + \beta_j + \gamma_{ij} + u\\
u \sim \mathcal{N}(0,\sigma^2)
$$
The total sum of squares can be decomposed as followings
($*$-index ...
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Chi-Square Test vs Distribution confusion
Disclaimer: I'm new to stats so please bear with me
Everywhere I look, the Chi-Square Distribution is explained with a (Z-Score)^2 i.e
$$ ((X-\mu )/\sigma )^2 $$ and based on a random variable from a ...
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How does Pearson's cumulative test statistic approach Chi-squared distribution?
From Wikipedia,
$ \sum_1^k{Z_i^2}$ is Chi-squared distributed($Z_i$ is a standard normal random variable)
Also, it is followed by that Pearson's cumulative test statistic $ \sum_1^n{(O_i-E_i)^2 \over ...
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Independence of $\chi^2$ distribution [closed]
Suppose we are given that $X_1$ and $X_2$ are two non-negative random variable such that $$X_1+X_2\sim\chi^2_{(2)}$$. Also we are given that $X_1\sim\chi^2_{(1)}$. Can we say the following
$X_2$ is ...
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Scalar product of Gaussian random vector with projection matrix is chi-squared
We define the $n$ chi-square random variable this way : if $Z \sim N(0,I_n)$ is multivariate Gaussian random vector, then $\lVert Z \rVert ^2 = \sum_{i=1}^n Z_i^2$ (sum of $n$ standard gaussian RV ...
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Degrees of freedom of Sample Variance of Residuals (Chi-Square distribution?
In the context of jointly testing J linear restrictions, I am reviewing the distribution of the F-statistic, which is F(J, n-k). Below, R is a full row rank Jxk matrix, q is a Jx1 vector, and the null ...
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Distribution of exponent of multivariate normal distribution
My slides say that the exponent of a multivariate normal distribution, $(\mathbf{X} - \boldsymbol{\mu})^\text{T} \boldsymbol{\Sigma}^{-1} (\mathbf{X} - \boldsymbol{\mu})$, follows a chi squared ...
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What are the relevant null distributions to infer the results of a simulation that renders a multinomial distribution with three possible results?
I have simulated 10 000 occurrences of a soccer match, by generating random numbers from 0 to 1, and then passing these, and the adjusted expected goals by team for the match, through an inverse ...
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Detecting multivariate outliers with Minimum covariance discriminant and mahalanobis distance
I've read in some papers (such as this) and CrossValidated questions (such as this, that people are using mahalanobis distance based on robust estimations of location and scatter using minimum ...
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How to prove $(\hat{X}-\mu)/(\hat{S}/\sqrt{n})$ is student t with $n-1$ degrees of freedom if $X_i$ are iid $N(\mu, \sigma)$?
It is commonly stated that if $X_i$ are iid $N(\mu, \sigma)$, then with $\hat{X}$ the sample mean, and $\hat{S}$ the sample error (sample standard deviation), then $\frac{ \hat{X}-\mu}{\hat{S}/\sqrt{n}...
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Degrees of freedom of a coin toss
The probability of a fair coin landing heads is $p(H)=1/2$ and since there is only one other outcome we can deduce that the probability for tails is also $p(T)=1-p(H)=1/2$. Yet if we examine a ...
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Is true that the sampling distribution of $\ln \left(\chi^{2}\right)$ converges to normality much faster than the sampling distribution of $\chi^{2}$?
If true is the consequence true that $X \sim \chi^{2}(k)$ then $\sqrt{2 X}$ is approximately normally distributed with mean $\sqrt{2 k-1}$ and unit variance?
Also true that If $X \sim \chi^{2}(k)$ ...
user318514
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Is it necessary to assume normal errors and independence in a linear regression model?
I am reading Casella Berger and in the book (pg. 545), they mention that it is possible to construct a linear regression model by only assuming that:
$$
E(Y_i) = \alpha+\beta X_i \qquad \text{with} \...
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Analytical Expression for Moment of Generalized chi-squared distribution
Consider $Z\sim N(0,1)$ and the moments:
$$
E\left[ \left(C_1(Z+\sqrt{\lambda})^2 - C_2\right)^t\right].
$$
Here, $C_1$, $\lambda$ and $C_2$ are arbitrary constants which are all positive. $t$ is a ...
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CDF of product of independent chi-square distributions
Assume that we have $i=1,\ldots,n$ independent chi-square distributed random variables,
$$
X_i \sim \chi^2(k_i).
$$
What is
$$
P\left(\prod_i X_i \leq c\right)
$$
for some $c>0$?
Note: I am hoping ...
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What is the distribution of the Frobenius distance between two covariance matrices?
I am computing the Frobenius norm of the difference between two covariance matrices,
\begin{align}
|\mathbf{C}-\mathbf{C}'|_F=\sqrt{\sum_{i,j}\left(c_{ij}-c'_{ij}\right)^2}.
\end{align}
Each of these ...
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