# 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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35 views

### Calculated Chi-Square probability by hand (using R) seems quite off - did I something wrong?

I try to understand the Chi-Square distribution with df=4-1. Therefore I tried to calculate the following (using R): I have a four-sided dice and I threw this dice n=100 times. I get the results (23,...
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### How is the chi-square confidence interval derived from the inverse gamma function?

I had to derive the chi-squared confidence intervals for a AR(1) red noise model generated theoretically to fit the power spectra of a time series. The shape function of the power spectra of the red ...
30 views

### How to measure variance of distances from origin

I'm trying to measure the sample variance of some data. Such data are 2D euclidean distances from the origin (0,0). Supposing to have the 2 components X and Y used to calculate the distance, it's ...
32 views

### Calculating p-value using chi-squared algorithm for an industrial process

I've had a problem assigned to me that ought to have gone to a statistician or process control expert, but it is what it is. Gist of the problem: I need to calculate a p-value from product yield data ...
39 views

### Let $X_1, X_2, … ,X_n$ be independent standard normal random variables. Find the distribution of the following random variables

I am studying for a preliminary examination in the fall over probability and have come across this question: Let $X_1, X_2, ... ,X_n$ be independent standard normal random variables. Find the ...
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### Averaging Gaussian Q function over Chi-squre distribution

Assume that we have $$P(\alpha) = Q(\sqrt{\alpha \rho})$$ where $Q(\alpha)$ is the Gaussian Q function defined as $$Q(x) = \frac{1}{\sqrt{2\pi}} \int_x^\infty \exp\{-\frac{u^2}{2}\} ~du$$ Can ...
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### Boundedness of a $\chi^2$ distribution with 1 degree of freedom

I am trying to see if this particular random variable related to $\chi^2$ distribution with $1$ degree of freedom (so standard normal RV squared) has a bound for all $n \in \mathbb{N}$. The random ...
14 views

### PDF for a chi-squared distribution divided by its degrees of freedom

What would the pdf be for a chi-squared distribution divided by its degrees of freedom. Would it be the normal pdf / degrees of freedom, or is it more complicated?
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### Chi-squared of two groups

I have to assess a claim that "the campaign to teach parents not to let their babies sleep on their stomachs is believed to have halved the number of deaths from SIDS in the last two years". I am only ...
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### Tail Bound for Squared Noncentered Gaussian

I am trying to upper bound the event $$P((x-\lambda)^2 < c)$$ where $x \sim N(\mu, 1)$, $c > 0$, and $\lambda \in [0, 1]$. While I am aware of chi-squared tail bounds for standard squared ...
30 views

### How to find the probability of one sample variance is two times larger than another?

I have two normal distribution where $X\sim N(\mu_{x}, 40^{2})$ and $Y\sim N(\mu_{y}, 50^{2})$. 8 samples from X and 16 samples from Y is drawn. How to determine the probability that the variance of ...
33 views

### How to calculate Chi-Square density value only known P-value?

Everywhere online there is how to calculate the Chi-Square density value given a confidence level: α/p value; but I can not find how one calculates the inverse? How to calculate the density value ...
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### Finding the value of a sample statistic using chi-squared distribution?

The problem is Find the variance $S^2$ for random sample of size 21 from a normal population with variance 5. (Hint: Use the fact that the statistic $\frac{(n-1)S^2}{\sigma^2}$ has a Chi-squared ...
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### Infinite sum of squared normal random variables with variances tending to zero

Let $Z_n \sim N(0, n^{-1})$ be independent. I would like to know what I can say about the distribution of $$Z := \sum_{n = 1}^{\infty} Z_n^2$$ Each $Z_n^2$ can be written as $\frac{Y_n}{n}$,where ...
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### concentration of function of Chi squared random variables

Let $X, Y$ be iid Chi-squared random variables with parameter $k$ and consider, \begin{align*} Z = \frac{X-Y}{X+Y}. \end{align*} I am after bounds for the tail: $\mathbb{P}[ |Z| > t ]$. I know the ...
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### approximating integral involving chi-squared

Is there a way of approximating the following integral $\int_{0}^{\infty}h(y)dF(y)$, where $F(y)$ is the chi-squared distribution with one degree of freedom (and $h(y)$ is a general integrable ...
25 views

### Carry out in $R$ the test of fit Pearson's chi-square to validate or invalidate the gamma model

A sample of size $n = 100$ includes: $14$ observations located between $0$ and $1$, $43$ observations between $1$ and $2$, $30$ observations between $2$ and $3$, and finally $13$ observations ...
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### (mis)Understanding the reduced $\chi^2$ statistic

In chapter 8 of the book "Measurements and their Uncertainties" by Huge and Hase, they discuss the reduced chi squared statistic. They define it as $$\chi^2_v = \frac{\chi_{\text{min}}^2}{v}$$ ...
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### On the pooled variance in the estimating difference of means

Suppose that we need to compare means of two sample spaces, say P, Q. Let $\mu$ and $\nu$ be the expectation value of $P$ and $Q$, respectively. Consider $m$ independent random samples from $P$ and $n$...
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### Chi-Square fitting - How to calculate the standard deviation of points in integrated functions?

For a Chi-Square fit one needs to determine the standard deviation $\sigma_{i}$ of each data point. $$\chi \equiv \sum_{i=1}^{N} \left(\frac{y_{i}-y}{\sigma_{i}}\right )^2$$ I have a derivative ...
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### What is the distribution of the following random variables, where the inputs are $N(0,1)$?

$X$ and $Y$ are independent $N(0,1)$ random variables: What is the distribution, of $$\frac{2XY}{\sqrt{X^{2}+Y^{2}}}?$$ I know the numerator is a difference of two independent $\chi^2$ random ...
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### Relations of $\chi^2$ random variable and normal random variable and answering certain problems

Correct me if I'm wrong. If $X$ is a random variable and if your trying to find the probability of $$p(X^2 > 1)$$ you would use the chi-square ($\chi^2$) table so if you have a problem asking ...
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### Sketching power function for a log normal density

Question: Hello, I am attempting all parts of the attached question. I have done part a, b and c. I have 2 questions. For part b) I am not quite sure what it means to "argue" that there is a ...
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### Probability between two values of a gamma distribution

I am trying to solve a problem where I have a random variable X that follows a gamma distribution of $\Gamma(\theta = 5, \alpha = 4)$. I am trying to find the probability of $P(10 \leq X \leq 30)$. ...
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### Determining covariance of two parameters given explicit relationship

I am wondering if it is possible to determine the covariance, $\text{Cov}(a,b)$, of two fitted parameters given I know their explicit relationship $a=a(b)$? I would like to construct the covariance ...
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### $\chi^2$ goodness of fit test: normal approximation and division of classes

When doing a $\chi^2$-test on deciding whether or not an RV follows a certain distribution, we perform $n$ observations of the RV, calculate the frequency that the observed value lies inside certain ...
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### Does Chi Square Square statistic always works for multinomial hypothesis testing?

The $𝒳^2$ statistic has been frequently employed in the hypothesis testing of the multinomial distributions. But after looking at the derivation of such a process(where they use Taylor expansion and ...
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### Show that distribution of function of LRT statistic for normal mean hypothesis testing is normally distributed

Suppose $X_1 ... X_n$ ~$^{iid}$ N($\mu, \sigma$), with $\sigma$ known. What is the distribution of $-2ln(\lambda)$ where $\lambda$ is the LRT statistic for testing $H_0:\mu = \mu_0, H_1:\mu \neq \mu_0$...
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### chi square test goodness of fit (frequency vs % frequency)

I'm trying to use the chi square test goodness of fit to see if heights in a survey are normally distributed. I'm using google sheets to do the calculations. When I look at the charts the distribution ...
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### Basic question about Chi-Square goodness of fit test

i am given a sequence of 20 random numbers. .05,.15,.23,.18,.06,.27,.31,.36,.32,.45,.55,.56,.51,.67,.62,.83,.96,.90,.70,.77. All in the interval [0,1) And now i am supposed to check, if those ...
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### Where in my derivation did it go wrong? Chi square statistics

Suppose $X_1, ..., X_n$ are iid normal with parameters $(\mu, \sigma^2).$ What would be the distribution of $\frac{\sum(X_i-\bar X)^{2}}{\sigma^2}?$ I think the right answer would be $\chi_{n-1}^2$, ...
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### Evaluate $\int_0^\infty x^{n+\frac12}e^{-\frac x2}\log^2x\,dx$ and $\int_0^\infty x^ne^{-\frac x2}\log^2x\,dx$

Determine the closed forms of $$\mathfrak I_1=\int_0^\infty x^{n+\frac12}e^{-\frac x2}\log^2x\,dx\quad\text{and}\quad\mathfrak I_2=\int_0^\infty x^ne^{-x/2}\log^2x\,dx$$ where $s>0$ is an integer. ...
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### Chi-squared distribution multiplied by constant

I have a question regarding notation. When $\chi^2(p)$ is the chi-squared distribution with degrees of freedom $p$, then what does it mean that $$X \sim k \chi^2(p),$$ where $k$ is some constant? ...
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### Is sample mean the optimal estimator of the mean of a random variable with Chi-Square distribution?

Assume there is a random variable $Y\sim \chi_k^2$, and $N$ independent realizations $y_i,i=1,...,N$, of $Y$. Is the sample mean $\hat{k}=\frac{1}{N}\sum_{n=1}^{N}y_i$ the optimal estimator of $k$ (...
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### Chi square substraction [duplicate]

I have a question regarding the substraction of chi-square distribution. If we have $X \sim {\chi}^2(n)$ and $Y \sim {\chi}^2(n)$ and X and Y are assumed to be independent, which distribtion does $X-Y$...
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### $\chi^2$ distribution: hypothesis testing

This is actually a statistic question from my econometrics class. It's not very difficult but I am having a lot of trouble with its wording. Suppose we have a sample of 10 independent observations, ...
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### Expectation of the ratio of two $\mathcal{X}^2(1)$ random variables

Let $X$ and $Y$ be two possibly dependent Chi-squared random variables both with $k=1$. Is it possible for $\mathbb{E}[X/Y]$ to be unbounded?
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### Finding percentiles of the $\chi^2$ distribution [closed]

If the random variable $Y$ has a $\chi^2$ distribution with $54$ degrees of freedom, then what is the approximate $84^{\text{th}}$ percentile of $Y$? I don't understand very good question when it is ...
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### Why does the Chi-squared test statistic follow the Chi-squared distribution

I know the Chi-squared test statistic is defined as: $$\chi^2=\sum_{i=1}^n\frac{({O_i-E_i})^2}{E_i}$$ where $O_i$ is observed data, and $E_i$ is expected. I also know that the $\chi^2$ distribution ...
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### Dot product of normal random vector with covariance matrix in between is chi squared distributed

I wish to prove that if $Y \sim N(0, \Omega)$, then $Y^T \Omega^{-1}Y \sim \chi^2_{k}$ where $k$ is the rank of $\Omega$. Any hint on how this can be done?
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### Chernoff analysis to prove lower bound, beefy hint provided

This question has some background that may be relevant, I'll try my best to be thorough but succinct. $A \in \mathbb R^{n \times n}$ is an unknown symmetric positive semidefinite matrix. Our goal is ...
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### Does $\sum_{i=1}^n Xi Yi$ for independent $X_i \sim \mathcal{N}(0,1)$ and $Y_i \sim \mathcal{N}(0,1)$ form a Chi-Square distributed Random Variable?

We all know that a chi-square distributed random variable: $U \sim \chi_n^2$ results from: $U = \sum_{i=1}^n X_i^2$, where $X_i \sim \mathcal{N}(0,1)$ and all $X_i$ are i.i.d. So, my question is ...
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### Can someone verify why the following alternate derivation regarding a non-central chi distribution giving different answer?

I am having trouble to understand why an alternate derivation of mine related to the Eq.(10) of this paper is not matching with the result in the paper. First, let me describe the problem. The paper ...
### Calculating $p$-value for a $\chi^2$-distribution?
The sugar content of the syrup in canned peaches is normally distributed. Suppose that the variance is thought to be $18 \text{mg}^2$. A random sample $n = 10$ of cans yields a sample standard ...