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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17 views

Multiplying a chi-square distribution by a constant

If $X\sim\chi^{2}(3)$. What is the distribution of $2X$?
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Can I use scipy.stats.chisquare and chi2_contigency interchangably?

The original question was here. Here is my extended question - If I use scipy.stats.chisquare, and set any one group as the "expected distribution" and the other group as "observed"...
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What are the mean and variance of the square of a chi square?

Let $x$ be a random gaussian variable with mean=0 and sd=1, which is then squared (thus a chi-squared variable), so $y=x^2$. I understand that the expected value of $y^2$ is actually the variance of $...
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38 views

Transformation of standard normal into chi-squared

I am trying to compute the marginal pdf of transformed standard normals. I'm not sure if I have followed the method correctly. Any help would be most appreciated. Let $X_1, X_2 \sim \mathcal{N}(0,1)$. ...
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Intuitive way to see how degrees of freedom affects the mean of a chi square distribution?

I am new to Statistics and trying to intuitively understand how a change in degrees of freedom affects the mean of a chi-square distribution. Suppose, We have $n$ normal random variables such that $...
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Determine independence/dependence of random variables

Let $Z_1$ and $Z_2$ be independent standard normal random variables. Let $W = \frac{Z_1 + Z_2}{\sqrt{2}}$ so that $W \sim N(0,1)$. Let $U = Z_1^2 + Z_2^2$ so that $U \sim \chi_2^2$. How can I ...
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Degress of Freedom in ANOVA with restrictions in parameters

I don't know how to find answer to this question. The answer is given as option (D). I know that total degrees of freedom is 18-1 = 17 and degrees of freedom for factor $\alpha $ and $\beta$ are 1 ...
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22 views

Does sample variance has a Chi-square distribution?

Let $X_1, X_2, \ldots, X_n$ be a random sample from $N(\mu, \sigma^2)$. Does $S^2=\frac{\sum^n_{i=1}(X_i-\bar X)^2}{n-1}$ has a Chi-square distribution? I know that $\frac{(n-1)S^2}{\sigma^2}=\frac{\...
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18 views

What is the limiting distribution of $\chi_r^2$ random variable, where $r\to 0^+$

What is the limiting distribution of $\chi_r^2$(Chi-square) random variable, where $r\to 0^+$. The following picture shows that as $r\to 0^+$ the distribution become degenerated in zero point. If it ...
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Finding functions of chi-squared or T distribution functions

I have no idea how to start this question. I'm not sure what happens when you divide a chi-squared variable by a constant for a). b) looks like chi-squared with degrees of freedom 3 and c) looks like ...
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27 views

Chi-squared and T distribution when s.d isn't 1

I have answered the first question but I have no clue where to start with b) and c). I'm pretty sure b) looks like the chi-squared distribution but am not sure how to work anything out as the standard ...
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Losing degrees of freedom for chi-square random variable

Let's say $Y=\alpha + \beta x + e$ is a normal random variable, with parameters $\alpha, \beta$ and a normal error $e$. When data points $(x_{i}, Y_{i})$ are taken $(i = 1, 2, 3, ... ,n)$, then the ...
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Transformation of dependent normally distributed random variables

If $Y_1,Y_2,...Y_n$ are normally distributed random variables with mean $E(Y_i)=\mu\;,Var(Y_i)=\sigma^2\;and\;Cov(Y_i,Y_j)=s[i,j=1,2,...,n;i\neq j]$ and we take the transformation $Z_i=Y_i^2$, then ...
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What is the distribution of $(X−Y)^2+(Z−Y)^2$, where $X$,$Y$ and $Z$ are independent normal distributions with their own means and variance? [duplicate]

I came up with a question: What is the distribution of $(X−Y)^2+(Z−Y)^2$, where $X$,$Y$ and $Z$ are independent normal distributions with their own means and variance? The common part is $Y$ in both ...
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Likelihood ratio test for nested model

I'm having a question about a likelihood ratio test in favor of the simpler, nested model. Assume we have a complex model $M_1=(\alpha, \beta)$, that correctly describes the data, and another, nested ...
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1answer
42 views

Expectation of inverse square under multivariate standard normal

In one of the steps in my lecture notes, the following result was used without proof: Given $X$ is a $p$-dimensional multivariate normal distribution, where $p\ge 3$, centred on zero, with covariance ...
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Calculate confidence interval for the population variance [duplicate]

Here is the problem: When cheching the Chi squared distribution table, the it seems like in the solution the denominators should be switched, because for .025 quantile the value is 13.844 and for ....
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63 views

Finding variance of the quotient of normal distribution and chi-squared distribution

Given that $Z\sim N(0,1), Y \sim \chi^2_{v}$, and assuming that $Z, Y$ are independent, we define $W=\frac{Z}{\sqrt{Y}}$. I aim to find $E(W)$ and $Var(W)$, with possible defining of $v$. Finding $E(W)...
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If $X \sim \mathcal{N}(\mu,\sigma^2)$, then how is $X^2$ distributed?

If $X \sim \mathcal{N}(0,\sigma^2)$, then $X^2$ is distributed according to a scaled chi-square distribution. If $X \sim \mathcal{N}(\mu,1)$, then $X^2$ is distributed according to a noncentral chi-...
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If $Y|X \sim \mathcal{N}(0,1)$ then is $Y^2|X \sim \chi^2(1)$?

Suppose we have random variables $X$ and $Y$ such that $Y|X \sim \mathcal{N}(0,1)$. Can we then say that $Y^2|X \sim \chi^2(1)$? If we can, then what about when $Y|X \sim \mathcal{N}(0,\sigma^2/4)$, ...
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The meaning of upper 100 alpha(th) percentile

Recently, I learned about the chi-square distribution. In my class, I was told about the upper $100\alpha^{th}$ percentile $\chi^{2}_{\alpha}(k)$ and given the following definition: $$P(X<\chi^{2}_{...
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Chi-squared distribution: Do I lack information?

The text of a problem from my book is: The area of houses $(x)$ expressed as $m^2,$ hence $y=x/10,$ follows Chisq(9). what's the percentage of houses below $30\,m^2?$ Don't I lack a location/...
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Question on how to sample randomly from the given distribution

I have following distribution which looks as follows: $P\left(\kappa| u,v,\lambda,y\right) \propto \kappa^{-\frac{n}{2}}exp\left\{-\cfrac{1}{2\kappa}\left[\epsilon + (u_1-u_2)^2 + (u_2-u_3)^2 \right] \...
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Question Regarding Derivation of the Chi-Square Distribution

I have been trying to derive the formula for $\chi^2$ distribution with $n-1$ degrees of freedom, but I am still having trouble. Assume $A$ is an orthogonal matrix with first row inputs $A_{1i}=n ^ {-...
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1answer
200 views

Inverse Gaussian chi square connection

The inverse Gaussian distribution $IG(\mu,\lambda)$ is associated with the density $$f(x;\mu,\lambda) = \sqrt{\frac{\lambda}{2\pi x^3}}\,\exp\left\{-\frac{\lambda(x-\mu)^2}{2\mu^2x}\right\}\qquad \...
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Distribution of the combination of two Chi-squared distributions [duplicate]

The random variable $A$ has a $\chi^2$ distribution with $p$ degrees of freedom. The random variable $B$ is independent of $A$ and has a $\chi^2$ distribution with $q$ degrees of freedom. Show that $(...
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Is it possible for a chi-square distribution to have unit variance?

I have a problem set from a professor that has me puzzling. Here's the problem: x and y are i.i.d Gaussian random variables with a mean of zero and unit variance. What is the mean and variance of |z|2,...
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Intuitive way to connect gamma and chi-squared distributions

I understand that a chi-squared distribution is a special case of the gamma distribution. However, I find claims of "the math just works out" to be an unhelpful in remembering or ...
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Show change of expression of sample variance and explain the distribution

Show that $$ \sum\left(Y_{i}-\mu\right)^{2} / \sigma^{2}=(n-1) S^{2} / \sigma^{2}+\left[(\bar{Y}-\mu)^{2} n / \sigma^{2}\right] $$ can be changed into a form $$ \frac{1}{\sigma^{2}} \widehat{S}_{1}=\...
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146 views

Use Chebyshev's inequality to find a lower bound of a Chi-Square Distribution

I'm trying to solve the following exercise but I'm not sure if what I'm doing is right. "Let $X$ be an r.v. distributed as $\chi_{40}^{2}$. Use Tchebichev’s inequality in order to find a lower ...
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79 views

sampling distribution of sample variance (normal distribution)

It is mentioned in Stats Textbook that for a random sample, of size n from a normal distribution , with known variance, the following statistic is having a chi-square distribution with n-1 degrees of ...
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39 views

Intuitive understanding of chi-squared variance

The chi-square distribution is a sampling distribution of normal variance. A chi-square distribution with $m$ degrees of freedom can be expressed as sum of squares of $m$ i.i.d standard normal ...
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1answer
68 views

product of asymptotic standard normal distribution

Suppose $Z_n\xrightarrow{d} Z \sim N(0,I_p)$, why $Z_n^TZ_n\xrightarrow{d}\chi^2_p$? I encounter this problem when we get the asymptotic distribution of the maximum likelihood estimator (MLE). Suppose ...
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28 views

Confidence intervals/ellipses when variables are correlated

I was looking to create a confidence ellipse for my X and Y variable in order to identify potential outliers. I'm new to this area so my understanding and use of this method may be wrong (please ...
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Distribution of the pooled variance in paired samples

Suppose a bivariate normal populations with means $\mu_1$ and $\mu_2$ and equal variance $\sigma^2$ but having a correlation of $\rho$. Taking a paired sample, it is possible to compute the pooled ...
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Impossible to get about 70% of height of Likelihood when I project the edge 1 sigma joint distribution on the 1D Likelihood

This post has been initially asked on maths.exchange but I didn't get any help, so I try to transfer it on this forum hoping someone could help me (I am going to delete the post on maths.exchange to ...
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1answer
241 views

Why Is A Squared Standard Normal Variable A Chi Square Variable

If for any $i \in \lbrace1,2,...n\rbrace$ where $Z_i \sim N(0,1)$ and all $Z_1, Z_2, ..., Z_n$ are independent of each other, why is it that $Z_i^2 \sim \chi^2_1$ and $\sum_i Z_i^2 \sim \chi^2_n$ ...
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GLMs with skewed distributions - why use mean and not mode?

There's something that's a bit troubling for me. The unit deviance in GLM is defined as $2[t(y,y) - t(y,\mu)]$, when $t(y,\mu) = y\theta(\mu) - b(\theta(\mu))$ (theta being the natural parameter). For ...
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1answer
150 views

Mean and variance of $Y'\Sigma^{-1}Y-Y_1^2/\sigma_1^2$ when $Y\sim N_2(0,\Sigma)$

Let $\underline Y=(Y_1,Y_2)'$ have the bivariate normal distribution $N_2(\underline0,\Sigma)$, where $$\Sigma=\begin{pmatrix}\sigma_1^2 & \rho\sigma_1\sigma_2 \\[1em] \rho\sigma_1\sigma_2 & \...
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1answer
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Interpretation for changes in a $\chi^2$'s density as $k$ increases

The chi-square's density becomes more regular as $k$ increases: $k=1$ unbounded, convex $k=2$ bounded, convex $k=3$ close to 0 near 0, unbounded positive slope $k=4$ close to 0 near 0, bounded ...
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Chi square approximation of the likelihood test ratio

I wasn't able to find any satisfying answer about that topic. I hope someone who understand correctly the subject could enlighten this shadow. This is not very important, just for the sake of ...
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1answer
73 views

Transformation of a random variable with a gamma distribution

Suppose $X_i \stackrel{i.i.d}{\sim}$ Exp$(1/\theta)$ which implies $\sum_{i =1}^{n} X_i \sim$ Gamma $(n, 1/\theta)$. But, then, the book that I am reading says that $(2/\theta)\sum_{i =1}^{n} X_i \...
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1answer
379 views

Square of Normal Distribution [duplicate]

I have a normal distribution $X$~$N(\mu,\sigma^2)$. Is there an exact value for the mean and standard deviation of $X^2$? Thanks
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1answer
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Kolmogorov-Smirnov interpretation in R for chi-square

With the sample size=500, I want to test whether the data follows chi-square distribution. For contiunous and one-dimensional distribution of the data, I use Kolmogorov-Smirnov test. Here I present ...
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1answer
107 views

When is uniform distribution have maximum entropy instead of normal distribution?

As far as I know, when we have just data and no constraints (other than probabilities must add up to 1), the distribution that gives maximum entropy is uniform distribution. But when we know mean and ...
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38 views

Variance, mean and understanding of the confidence interval for chi-square?

For some provided data, I have to compute confidence interval for variance and mean. As I look for quantile plot, it seems for me that the data distribution follows rather chi-square distribution than ...
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Saddlepoint approximation of the generalized chi-square distribution

Following the discussion found here and here, I have been trying to derive the saddlepoint approximation for the generalized chi-square distribution, with the moment generating function defined in ...
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1answer
134 views

How can I show that $\frac{1}{\sigma^2}\sum^k_{i=1}n_i[(\bar{Y}_{i.}-\bar{\bar{Y}})-(\theta_i-\bar{\theta)}]^2 \sim \chi^2_{k-1}$?

Define $\bar{\bar{Y}}=\sum n_i \bar{Y}_{i.}/\sum n_i$ and $\bar{\theta}=\sum n_i\theta_i / \sum n_i$, where $Y_i \sim N(\theta,\sigma^2)$. How does can I show that $\frac{1}{\sigma^2}\sum^k_{i=1}n_i[(\...
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1answer
30 views

What exactly is a inverse $\chi^2$ distribution?

What exactly is a inverse-chi square distribution?
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48 views

Independence of MLEs of 2 parameter exponential, and showing functions of them are chi-square

Consider a random sample of size $n$ from a two-parameter exponential distribution, $X_i \sim $EXP($\theta,\eta$), and let $\eta^*$ and $\theta^*$ be the MLEs. a) Show that $\eta^*$ and $\theta^*$ are ...

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