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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I have to determine $d_{\alpha}$ in terms of critical values of a chi squared disdtribution, significance level $\alpha$ and sample size n

So far I have the following: Assuming the Null hypothesis $X_{i} \thicksim$ EXP$(1)$ $Y: = \sum_{i = 1}^{n} X_{i}$ $Y \thicksim$ GAM$(1,n)$ $\bar{X} = Y/n$ $P(\bar{X} \leq$ $d_\alpha) = 1 - \alpha$ $P(...
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What is the distribution of a quadratic function of normal distributions? [closed]

Suppose $Z_i$ is independent standard normal distributions, i.e. $Z_i\sim N(0,1)$, $i=1,2,\cdots, d$. What is the distribution of $$ \sum_{i=1}^d (a_iZ_i+b_iZ_i^2). $$ I know when $a_i=0$, it is the ...
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Why can a $\chi^2$ test be used to check whether a random variable has any distribution over a finite space?

In particular, I see it used to test whether a random variable is Uniform even though the graph of the $\chi^2$ pdf doesn't ever seem to look much like a uniform distribution's pdf, and I don't ...
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$L^2$ Approximation error in Gaussian Process Regression (finite data setting)

I am learning about Gaussian Process Regression. I would like to have some references or results regarding the distribution of the error between a given function, and the posterior obtained in ...
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How to get Information Entropy and Chi-Square Test value for an encrypted image

I was reading this paper. Under the section 4.3 and 4.5 they add The Chi-Square Test Analysis of Cipher Image and Information ...
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in-probability bound for maximum of chi square distributions

Let $X_1, \dots, X_n$ be i.i.d. Chi square random variables with $d$ degrees of freedom, i.e. $X_1, \dots, X_n \stackrel{\text{i.i.d}}{\sim} \chi^2_d$. There is a nice bound (see Example 2.7 from the ...
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Choosing the correct t test for comparing two sample means

Six sets of identical twins were divided at random into two groups, each group containing one twin from each set. The first group was taught some basic statistics by method A and the second by method ...
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Estimating probability of $\|Z_1\|^2-\|Z_2\|^2$ in a bivariate normal population

Suppose $(Z_1,Z_2)\sim N_2(\mu,0; 2,2; \frac{1}{2})$, i.e., a bivariate normal distribution with mean vector $(\mu,0)$ with each component having variance $2$ and covariance between the variables ...
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Can I use the Chi Squared Test on percentages?

I am writing a program where I want to check the fit of some data to a distribution through the Chi Squared test. The data is relative frequency, so could I use the percentages in each cell from the ...
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Pearson's Chi squared goodness of fit test for averaged statistical distribution

Suppose if I have one single measurement of a statistical distribution binned $p(x)$ vs $x$, I could apply a Pearson's Test to calculate $\chi^2 = (O_i−E_i)^2/E_i$. Then I would compare the critical ...
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Why Null Hypothesis for contingency tables is 'independent'?

For contingency tables I don't understand why the Null Hypothesis is 'independent'. We calculate Sum(Observed^2 / Expected) - N and compare this with the chi-squared distribution table. Let's say: Sum(...
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Chi-square and Taylor expansion of relative entropy

I'm trying to show a relation between relative entropy and Chi-square, more specifically,$\chi^2 = \sum_{x}\frac{(p(x) - q(x))^2}{q(x)}$ is twice the first term in the taylor series expansion of $D(p||...
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reading my p-value from chi-square vs my polt data

I'm a bit confused with the result I get and wonder if someone can put light into my understanding. consider the following table (neighborhood vs education level): ...
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Why 2 moment generating functions are multiplied each other in equation of chi-squared distribution?

$$X_{1},\ldots,X_{n}~\leftarrow~~\text{random samples from}~~\mathcal N(\mu_{},\sigma^2)\tag{1}$$ $$\bar X={1\over n}\sum_{i=1}^{n}X_{i}\tag{2}$$ I want to prove the following. $$\sum_{i=1}^{n}\left({...
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Why$~\sum_{i=1}^{n}\left({X_{i}-\bar X\over\sigma}\right)^2\sim\chi_{n-1}^2~$can be held in chi-squared distribution?

$$X_{i}:=\text{ith sample which was extracted from}~~\mathcal{N}(\mu,\sigma^2)\tag{1}$$ $$i\in\mathbb{N}_{n}^{*}~~\Leftrightarrow~~i\in\{1,2,\ldots,n\}\tag{2}$$ I assume these random variables are ...
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$X,Y$ are independent stanard normal variables. What is $E[\frac{X^2}{Y^2}]$?

If $X,Y$ are independent standard normal variables, then $X^2, Y^2$ are independent chi-squared r.v with 1 degree of freedom, i.e. $X^2 \in \chi^2_1, Y^2 \in \chi^2_1$. I am trying to find the mean ...
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Taylor Expansion of F-critical value around Chi-square critical value

I am looking at a proof that takes a Taylor series expansion around a (1 - $\alpha$) critical value $\mathscr{F}^\alpha$ for F-distribution $F_{p,K}$ around the corresponding Chi-square critical ...
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How to prove whether $T_3$ has a chisquare distribution subject to a multiplicative constant

Suppose $X_1,...,X_n$ are random samples from $N(0, \sigma^2)$, and $\bar X_n = n^{-1}\sum_{i=1}^{n}X_i$. Let $Y =$ $Y_1 \choose {Y_2}$, where $Y_1 = X_1 - \bar X_n$, $Y_2 = X_2 - \bar X_n$, be a ...
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Can't reproduce approximately the chi-squared test value, did I misconduct?

I want to reproduce the chi-squared test for goodness of fit as attached below the table. Deposits Actual frequency Negative binomial frequency Poisson frequency 0 8586 8584.26 8508.53 1 176 176.84 ...
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What went wrong in my application of the Martingale Central Limit Theorem to a degenerate U-statistic?

Suppose we have $X_1,\ldots,X_n\stackrel{{\rm i.i.d.}}\sim N(0,1)$, then we have $$ \dfrac{(X_1+\cdots+X_n)^2}{n} = \big(\sqrt{n} \bar{X}\big)^2 \sim \chi^2_1. $$ Applying law of large numbers to the ...
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Defining a chi-square distribution with zero degrees of freedom

Give a reasonable definition of a chi-square distribution with zero degrees of freedom. My textbook offers a hint to use the moment generating function (m.g.f.) of a distribution that is $\mathcal X^...
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Chi-square testing if distribution is uniform

I have a question around the Chi-Square test which I'm trying to determine if a given set of values has uniform distribution. I care about whether or not it can be shown with statistical significace ...
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How to convert Gamma distribution into Chi square distribution?

I have to find the distribution of the r.v. $-\sum_i^{10} log(x_i)$ and present it in the form of $\chi^2$ distribution. Given that: $-log(x_i) \sim Exp(\theta) $. Then I know that the sum of ...
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How to use Chi Square to compare attributes of sub populations

I have a population that can be subdivided into sub-populations (e.g., a sack full of Apples,Pears,Oranges). Every item shares a binary attribute (e.g.,ripe:true/false). I want to figure out if one of ...
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Area Under The Curve of CDF and Alpha Value - Chi-Sq Test Interpretation

Can you help me understand the meaning of the area under the curve of a PDF in the interpretation of a Chi-Square test? This is what I think I know: If my test statistic (i.e. the chi-square value) ...
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How to show that $|X|^2$ has the chi-squared distributions with r degrees of freedom?

The question is Suppose that the random vector $X$ has a centered k-dimensional normal distribution whose covariance matrix has 1 as an eigenvalue of multiplicity $r$ and $0$ as an eigenvalue of ...
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Why should the sum of squares of two Independent normals be memory-less

In section 11.3.1 of Introduction to probability models by Ross (10th edition), a very strange phenomenon is described. If you take two independent standard normal distributions and sum their squares, ...
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What is the mathematical derivation for the chi-square homogeneity test formula?

If we have J multinomial mutually independent random variables $N_j = (N_{1j},...,N_{Ij}) \sim Mult(n_j, \;p_{1j},...,\;p_{I-1,j}), \; \; j= 1,...,J$, and the test's hypotheses are $H_0: \forall i \...
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Chi square test for other cases apart from checking distribution and independence

I read one paper and the authors used the Chi-square test. But I have never seen that Chi-square is used for such case. Can someone share brief information for such kinds of cases? Full paper here ...
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computing cdf $\chi^2_3$ in Matlab

I have still another problem with chi2 in Matlab: How can I compute the vector c.d.f. This doesn't work chi2([0:0.01:1],3) ? ...
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When comparing MGFs why is $X+2Y \ne X+Y+Y$?

As the title states. I have $X \sim ChiSquare(2)$ and $Y \sim ChiSquare(2)$, i.e. they are both distributed as chi-squares with 2 degrees of freedom and are independent. Define a new random variable $...
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Comparison of bird diet at two different nests

I have a list with the different prey types and quantities that birds at Nest $1$ gave to their fledglings. I also have the same information for another nest of the same bird species. Suppose I have <...
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Where's my mistake in my attempt at showing that the squared sum of normally distributed variables is a $\chi^2$ distribution?

$\newcommand{\N}{\mathcal{N}}\newcommand{\d}{\,\mathrm{d}}$I am trying to understand how the sum of squares of standard normal variables is a $\chi^2$ distribution. Note that I am not a student of ...
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Integral-sum conversion in the proof that the $\chi^2$ statistic tends to the $\chi^2$ distribution

I cite the derivation on Wikipedia, here. The focus of that section of the article is to show that the $\chi_p^2$ statistic is asymptotically equivalent to the $\chi^2$ distribution. Let $n$ be the ...
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Limiting distribution of $W_n = Z_n/n^2$ where $Z_n\sim \chi^{2}(n).$ using pdfs

The goal is to find the limiting distribution of $W_n = Z_n/n^2$ where $Z_n\sim \chi^{2}(n).$ I know from this answer that $W_n\rightarrow 0$ in distribution. I am trying to get this answer using CDFs....
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Sampling distribution of chi-square random variable

Consider $W=[W_1, W_2, \dots, W_N]$, where each element is complex Gaussian random variables, i.e., $W_i = X_i+jY_i$ with $X_i, Y_i \sim \mathcal{N}(0, \sigma^2)$. Now, I am trying to find the ...
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How to verify chi-square random variables are sub-exponential variables?

I just began to study statistics this term and I had some trouble understanding the derivation of sub-exponential variables. Let $Z \sim \mathcal{N}(0, 1), X = Z^2 \sim \chi^{2}_{1}$. Now, I want to ...
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Find E[e^(aX^4)], where X ~ N(0, sigma^2)

I have to compute $\mathbb{E}\{e^\left(aX^4\right)\}$, where $X \sim \mathcal{N}(0, \sigma^2)$ is a normal random variable. Following the top answer here, I tried to find the solution. This requires ...
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The intuition behind the chi-square distribution or the square of a random variable

Lets call X~N(0,1) I can't understand why squaring normal random variable outputs a chi-square variable with one degree of freedom. The problem might be I ignore the intuition behind de the ...
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Deriving the expected value equation in chi-squared tests.

I'm struggling to get my head around a statistics idea (I study medicine). In calculating chi squared, the expected value is first calculated by using the following equation: $\frac{\text{row total}\...
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How to show that likelihood ratio test statistic for exponential distributions' rate parameter $\lambda$ has $\chi^2$ distribution with 1 df?

Let $X_1, X_2, ..., X_n$ be iid. random variables with pdf. exponential. Let $H_0 = \theta = \theta_0 $ and $H_1 = \theta = \theta_1$. Then test statistic for likelihood ratio would be: $$ LR = \frac{...
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3 votes
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Multivariate Generalized Chi-square distribution

A generalized chi-squared distributed random variable $\xi$ can be expressed as $$\xi=q(x)=x'Q_2x+q_1'x+q_0,$$ where $x\sim\mathcal{N}\left(\mu, \Sigma\right)$, $Q_2$ is a matrix, $q_1$ is a vector ...
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Finding and identifying the distribution of the sample variance

Given a random sample of size $n$, from a normal population with variance $\sigma^2$, I am trying to find and identify the distribution of the sample variance, $S^2$. I know that $\frac{(n-1)S^2}{\...
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Finding the minimum sample size for the variance to lie within an interval

Suppose $X_1,X_2,...,X_n$ are $i.i.d$ normal with variance $\sigma^2$. We estimate its variance using $S^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\bar{X})^2$. What is the minimal integer n so that we are 95% ...
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Can we obtain the monotonicity of $F(v)=C_v\left[aC_v^{-1}(b)+c\left(v-\sqrt{2v}\right)\right]$ with $v$?

Suppose we have a function: $F(v)=C_v\left[aC_v^{-1}(b)+c\left(v-\sqrt{2v}\right)\right]$ where $v$ is a positive integer, $a,b,c$ are constants and satisfy $a,c>0$, and $0< b <1$. $C_v(\cdot)...
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Logarithm of normal distribution is constant plus Chi squared?

I'm reading Bayesian data analysis. On page 85 it is stated that "In the d dimensional normal distribution, the logarithm of the density function is a constant plus a $χ^2_d$ distribution divided ...
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Distribution of normalised chi-square distribution

I'm doing a data analysis where the data is right-skewed, similar to a $\chi^2$-distribution. The data is then normalised to their z-scores. $z = \frac{X-\mu}{\sigma}$ I want to do an outlier analysis ...
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Expectation of logarithm function with chi-square distributed variables

How to calculate $\mathbb{E}\left[\log\left(1+x\right)\right]$ where $x$ denotes a central chi-square distributed variable? The lower bound and upper bound of this expectation are also of my concern.
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Finding P($X^2$<1) , after verifying that $X^2$ is distributed $\chi^2_1$

The Problem Suppose $X$ and $Y$ are independent n(0,1) random variables. Find $P(X^2 < 1)$, after verifying that $X^2$ is distributed $\chi^2_1$. My Work I understand that I need to find the ...
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Why is the chi-square distribution skewed to the right and why does it only range from $0$ to $\infty$?

My book says that the chi-squared distribution is continuous, skewed to the right, and ranges from $0$ to $\infty$ with no explanation what-so-ever. It just gives me a table to tell me how to find $P(\...
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