# 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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1 vote
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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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1 vote
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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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### 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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1 vote
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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.
1 vote
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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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### 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 $X \sim \chi^{2}(k)$ ...
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