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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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Test goodness of fit for weighting samples

There are N samples $x_1, x_2, ...,x_N$ for random variable $X$. I know Pearson's chi-squared test can be used to figure whether X follows uniform distribution. First, discretize the range of $X$ ...
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Variance of linear combination

This is a follow up question to this. Let $(X_1,\ldots, X_n)$ be non-independent random variables such that $$\sum_{i=1}^{n} X_i\sim\sum_{i=1}^{n} \alpha (\mathcal{N}(0,1))^2$$ where $\mathcal{N}(0,1)...
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Why isn't the chi-squared test appropriate in this case?

I was trying to solve the following exercise: Two researchers studied the relationship between infant mortality and environmental conditions in Dauphin County, Pennsylvania. As a part of the study, ...
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40 views

Distribution of linear combination

Let $(X_1,\ldots, X_n)$ be non-independent random variables such that $$\sum_{i=1}^{n} X_i\sim\sum_{i=1}^{n} \alpha (\mathcal{N}(0,1))^2$$ where $\mathcal{N}(0,1)$ is a standard-normal distribution (...
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20 views

What are the “degrees of freedom” in this Chi Squared test?

I have learnt that the degrees of freedom are the (number of rows - 1) multiplied by (the number of columns - 1). However, I am stuck as to what the degrees of freedom are in the following set-up for ...
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Chi-squared analysis of an opinion poll

In an election the votes were distributed as follows: A: 26.3%, B: 4.6%, C: 3.4%, D: 0%, E: 0%, F: 4.2%, I: 7.5%, K: 0.8%, O: 21.1%, V: 19.5%, Ø: 7,8%, Å: 4,8% A recent opinion poll suggests the ...
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23 views

Calculate expected value and variance of a t-student distribution without using density function

Let $(X_n)_n$ a suite of random variables independent and identically distributed, $X_i \sim \mathcal{N}(0,1)$ and let $Y_n:= \sum_{j=1}^n X_j ^2 \sim \chi^2_n$ a chi-square random variable with $n$ ...
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14 views

Chi squared test - free parameters

I know that for a Chi squared test where we have $\Theta_0$ free parameters under the null hypothesis $H_0$ and $\Theta_1$ free parameters under the alternative hypothesis $H_1$, that $$2\log\Lambda \...
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1answer
38 views

Hypothesis Test when unknown mean and unknown variance

let have a random variable X which follows a normal distribution of mean $\mu$ and variance $\sigma$. we want to carry out the following hypothesis test: $H_0: \mu=\mu_0$ against $H_1: \mu\neq\mu_0$ ...
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Expressing quadratic form of normal variables in terms of chi-squared variables

On Wikpedia and in the references therein [1,2], it is stated that any quadratic form of normally distributed random variables can be expressed as the sum of many independent non-central chi-squared ...
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Chi-squared confidence intervals for big measurements

Let's say I have some observational data. For example: in a certain wavelength interval I measure the flux, 10 000 points, and I have the uncertainty for each one of them. For example: ...
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35 views

Derivation of expected value of $\exp(-aX^2)$ when $X$ is normally distributed

I am trying to derive the following result: $$\mathbb{E}\left[\exp\left(-aX^{2}\right)\right],$$ such that $$X\sim\mathcal{N}\left(\mu,\sigma^{2}\right),$$ $$a = a_{R} + ia_{I},$$ $$a_{R},a_{I} \in \...
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36 views

Chi square goodness of fit test for Exp(1) in r

Our teacher has posted the following task: We have 100 random numbers in our disposal that come from Exp(1) and the task is to perform the chi square test in order to check if the numbers come indeed ...
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40 views

Question about Chi Square distribution involving estimated variance

So I've come across this question (Exercise 17) whilst attempting some Probability/ Statistics questions, and I'm a bit confused about how to approach it. It firstly asks me a show that question, that ...
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Statistical Tests for Preferential Treatment in Government Contracts

I have a dataset of the contracts (tenders) issued by different government contracting authorities and the corresponding suppliers who were awarded the contract. I also know the date on which the ...
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Degree of freedom for $\chi^2$ test

First of all, I know there are countless questions on this topic already. But I still can't decide the degree of freedom for my table. This is my table: ...
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Bound on probability of bi-variate chi-square distribtuion

If I have two random variables $u_1 = x_1^2$ and $u_2 = x_2^2$ where $x_1,x_2 \sim \mathcal{N}(0,1)$ and $E(x_1x_2) = \rho$. The what is the pdf or bound on the pdf of $P(u_1<c,u_2>c)$? Edit: ...
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55 views

How to test if data diverge significant from exponential distribution using the Chi-squared test

How can I test, if the following data diverge from the exponential distribution with $\tau = 2.197$? The data has the form: $0 \leq x \leq 0.5 :$ 194 $ 0.5 \leq x \leq 1 :$ 117 $1 \leq x\leq 1.5 :$...
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Chi square comparing data sets

I am trying to use a chi square calculation to compare whether or not two data sets are similar. Suppose I have two sets of data, x and y, each containing 20 values, and each value has an ...
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23 views

Is there such a thing as scaled chi distribution?

I have $n$ variables normally distributed: $X_i$ ~ $N(\mu_i, \sigma_i^2)$ ($i=1,...,n$), independent. I am interested in the distribution of $$ Y := \sqrt{\sum \limits_{i=1}^n X_i^2}. $$ Is there any ...
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Mean of the Chi Distribution

I'm trying to find an online derivation of why the mean of the Chi-Distribution with $n$ degrees of freedom is $$E(\chi_n)= \sqrt2 \frac{ \Gamma(\frac{n+1}{2})}{\Gamma(\frac{n}{2})}$$ Is it too ...
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MVUE for variance of normal distribution

Q. We are given $Y_1,Y_2,...Y_n$ as NID(0,$\sigma^2$). Does there exist any unbiased eastimator for $\sigma^2$ whose variance follows the CRLB for $\sigma^2$. Considering $\tau(\sigma) = \sigma^2$. ...
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Homework about the norm of normal random variable

This is a homework, I really don't know how to start, I appreciate any hint. Let $X \sim \mathcal{CN}(0,\sigma_x^2)$, $Y \sim \mathcal{CN}(0,\sigma_y^2)$, $Z \sim \mathcal{CN}(0,\sigma_z^2)$ be ...
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Chi-squared distribution problem

Let $X_{1},X_{2},\cdots,X_{n_{1}}$ and $Y_{1},Y_{2},\cdots,Y_{n_{2}}$ denote independent random sample from the normal distributions $N(\mu_{1},\sigma^2_{1})$ and $N(\mu_{2},\sigma_{2}^{2})$, ...
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Additive property of independent chi-squares

in a user study the participants were shown 9 different visual representations of data and they had to answer (a) if they do understand the presented data and (b) if they think it is helpful. The ...
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55 views

The lower bound of the expected maximum of $n$ i.i.d. chi-squared random variables.

Let $X_1$, $X_2$, $X_3$, $\dots$, $X_n$, be i.i.d. chi-squared variables with the degrees of freedom $d$. I am struggling in finding the lower bound of $E[\max_{1\leq i\leq n} X_i]$. I guess its lower ...
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Confusion regarding usage of Mahalanobis distance for update rejection in Kalman filtering

I recently came across some material that discussed a method for performing update rejection in Kalman filters when bad measurements are received. [Paper 1] [Paper 2: see Section III(E)] This method ...
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1answer
105 views

Combining chi square charecterisitc

I am reading Statistics by David Freedman Suppose the same die had been used to generate the data in tables 4A and 4C (p. 531), rolling it first 60 times for table 4A, and then 600 times for ...
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Distribution of a squared standard Brownian motion

During a self study I encountered the following issue: I got a standard Brownian motion $B(t)$ on the interval $[0,1]$ which is $N(0,t)$ distributed by definition, but now I want to figure out the ...
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1answer
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Manual calculation of p-value from chi square and df

I am trying to create an algorithm that could calculate the p-value given the chi-square statistic and the degrees of freedom. I found the following formula here Can anyone please point me in the ...
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how to prove chi-square statistics conforms to chi-square distribution with contingency table?

chi-square test(principle used in C4.5's CVP Pruning), also called chi-square statistics, also called chi-square goodness-of fit How to prove $\sum_{i=1}^{i=r}\sum_{j=1}^{j=c}\frac{(x_{ij}-E_{ij} )...
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What is an intuitive explanation for how the t-distribution, normal distribution, F-distribution and Chi-square distribution relate to each other?

What is an intuitive explanation for how the t-distribution, normal distribution, F-distribution, and Chi-square distribution relate to each other? Could anyone explain this clearly with a sensible ...
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Making chi-square distribution out of unknown CDF [duplicate]

Suppose that the random variables X1,..., Xn are independent, and each random variable Xi has a continuous c.d.f. Fi. Also, let the random variable Y be defined by the relation Y = −2∑(i=1~n) log Fi(...
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Deriving the Chi-squared distribution using characteristic functions

I would like to directly derive the probability density function (PDF) for a Chi-squared distribution with $k$ degrees of freedom using characteristic functions. If $X_{1}, X_{2}, \dots, X_{k}$ are ...
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Mean and Variance of ${S ^ 2}$ when $\frac{(n-p)}{ \sigma ^ {2} } \cdot {S ^ 2} \longrightarrow {\chi ^ 2 }{(n-p)}$

I am looking for the detailled proof for this case, if $$\frac{(n-p)}{ \sigma ^ {2} } \cdot {S ^ 2} \longrightarrow {\chi ^ 2 }{(n-p)}$$ Prove that : $$E[{S ^ 2}] = \sigma ^ {2}$$ $$Var[{S ^ 2}] =...
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110 views

Showing that the difference of Chi squared random variables follows a chi squared distribution

how can I show that given $ \textbf{y} =(y_1,y_2)^T \sim N (\textbf {0} ,\Sigma ) $ $$(\textbf{y}^t\Sigma^{-1}\textbf{y} - \frac{y_1^2}{\sigma_{11}}) \sim \chi^2_1$$ where $ \Sigma =(\sigma_{i,j})$ ...
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Critical region for likelihood ratio test at level-$\alpha_0$

I am stuck at an exercise on a specific likelihood ratio test, I cannot figure out how the size of the test would be $\chi^2_1$ distributed. Problem description We consider a sample $X_1,...,X_n$ ...
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159 views

Uniform distribution, chi square test

What test or procedure can I use to determine the best estimate $\alpha\in [0,1]$ whether given $N$ numbers come from the uniform distribution in the interval $[0,\theta]$ for a given $\theta>0$? I'...
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Determining confidence intervals and confidence band for a basic polynomial regression problem

I'm working through Elements of Statistical Learning. I'm trying Exercise 3.2 (page 94 in the textbook hardcopy / page 113 in the textbook PDF), a polynomial regression problem which asks me to: make ...
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Self study-Distribution of quadratic form when dispersion matrix is positive semi definite

Suppose that $Y\sim N_n(\mathbf 0,\Sigma)$ and $A$ is a symmetric matrix. Then $Y'AY$ follows $\chi^2$ distribution with $r$ degrees of freedom if and only if $r$ of the characteristic roots of $A\...
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Proof of test statistic of Jarque-Bera test result from normal distribution are $\chi^2_2$

The test statistic of Jarque-Bera test are as follow $T=\left(\frac{\hat{S}}{\sqrt{6/n}}\right)^2+\left(\frac{\hat{K}-3}{\sqrt{24/n}}\right)^2$ While $\hat{S}$ is the sample estimate of skewness, $\...
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Find the distribution of the error

Suppose we want to find a circle to fit the data points. The error of the data point can be calculated as $e_k=||\vec{x}_k-\vec{x}_c||^2-r^2$ ------------------(1) where the $\vec{x}_k$ is the ...
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Optimal decision rules given a set of bimodal values

Some patients have two possible illnesses ; to determine which illness a patient has, we make a test (only once) that will result in one of five outcomes, which have the following conditional ...
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1answer
38 views

Independence of $V+W$ and $\frac{V}{V+W}$ when V is Chi square of degree 1 and W is of degree 2

I am trying to understand why $V+W$ is independent from $\frac{V}{V+W}$. With $V \sim \chi^2_{(1)}$ and $W \sim \chi^2_{(2)}$. I do not see how this comes about. Note: $V$ and $W$ are independent I ...
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1answer
55 views

$X^2$, when $X\sim N(0,1)$ : $2\frac{\operatorname{d}F_X(\sqrt{y})}{\operatorname{d} y}$

I was reading this: Characteristic function of Normal random variable squared And there is a step, where I do not manage to understand the equality marked as "$\stackrel{???}{=}$" We have $$ F_X\...
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Can we conclude that the advertising technique has an impact on sales?

The impact of three different advertising techniques is being studied by a marketing firm. Sales, in thousand dollars, categorized in four groups are shown for each advertising technique for 200 ...
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58 views

$\chi^2$-test for smoothing splines: degrees of freedom

Suppose we are given raw data, e.g. raw mortality rates $\widetilde{q_x}$, which are graduated by a smoothing (cubic natural) spline $S$. That is, we obtain smoothed rates by setting $q_x:=S(x).$ Let ...
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29 views

sub-exponential property of a linear transformation of independent random variables

I need to use a Bernstein-type bound to a random variable $y^Ty$, where $y^Ty= Qx$ is a linear transformation of independent variables $x$. I don't consider $y$ to be an independent variable. So to ...
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Changing the null hypothesis: Chi-Square test for Independency/ Homogeneity on a g x 2 table

My Goal is summarize the prerson statistic (Goodness-of-fit, Homogeneity and Independency). I want to know how and why the test works. For this I've read a lot of papers, including the original paper ...
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30 views

How to proper determinate the number of degrees of freedom using the Pearson Chi-Square test?

Introduction and notation My Goal is summarize the prerson statistic (Goodness-of-fit, Homogeneity and Independency). I want to know how and why the test works. For this I've read a lot of papers, ...