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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Why does one compute an outer product of marginal distributions (of contingency table) instead of splitting data up completely equally?

I am studying about the chi-squared test statistic and came across the following code. ...
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SOLVED: What is the distribution of the squared exponential of two gaussian distributions of odd dimension?

Let $X,X'$ follow the $n$ dimensional Gaussian distribution (zero mean unit variance for simplicity). My question is: what is the distribution of $Y = e^{-\|X-X'\|^2}$, particuarly when $n$ is odd? $...
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Distribution of $\mathbf{v}^{\top} \Sigma^{- 1} \mathbf{v}$, when $\mathbf{v}$ is a multivariate normal with covariance $\Sigma$? [duplicate]

What is the distribution of the quadratic form $\mathbf{v}^{\top} \Sigma^{-1} \mathbf{v}$, when $\mathbf{v}$ is a multivariate normal with covariance $\Sigma$ and zero means? I suspect this is related ...
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Moments (mean and skewness) of an AR(1) process with Chi2 or Gamma innovation distribution

A bit of context I am looking for a lag-1 autoregressive process with non-Gaussian innovation/residual error, which is capable of producing both skewed and non-skewed marginal distributions. I am ...
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How can we centralise a non-central chi squared random variable?

Say that $X\sim {\chi '}_{k}^{2}(\lambda)$ and $Y \sim \chi_k$. What transformation of $X$ will produce $Y$? If we also let $Z \sim N(\mu, 1)$, $\lambda = \mu^2$, and $k=1$, then I understand that $(Z ...
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How to find the Chisq values for a two tailed test in R? [duplicate]

How do we find the chisq value for a two tailed test? For left tail we use qchisq(x,df,lower.tail="True") For right tail we use qchisq(x,df,lower.tail="False") What about two ...
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Linear Regression, the distribution of SSE over sigma squared

How can one prove that $ \frac{SSE}{\sigma ^ 2} $follows a $ \chi_{n-p} ^ 2 $ distribution using matrix notation? Where n is the number of observations and p is the number of parameters in the model.
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Understanding the relationship between Chi-squared distribution and test statistic

Chi-squared distribution with $k$ degrees of freedom is defined as the distribution of the sum of the squares of $k$ standard normal random variables: $$\chi^2 = \sum_{i=1}^k Z_i^2$$ Where each $Z_i\...
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Reasoning for the DOF of $\frac{1}{\sigma^2} \sum_{i = 1}^n (Y_i - \mu)^2 \sim \chi_n^2$ and $\frac{1}{\sigma^2} n(\bar{Y} - \mu)^2 \sim \chi_1^2$?

I have the following example: Let $Y_1, \dots, Y_n$ be an i.i.d. $N(\mu, \sigma^2)$. Note that $\sum_{i = 1}^n (y_i - \mu)^2 = \sum_{i = 1}^n (y_i - \bar{y})^2 + n(\bar{y} - \mu)^2$. We show that $\...
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How is this implied by the properties of the exponential, gamma, and $\chi^2$ distributions?

Let's say we have the random variables $X_1, \dots, X_p$. Furthermore, say that these random variables are a random sample from a PDF of the form $$f_\tau (x) = \begin{cases} \tau x^{\tau-1}, & 0 ...
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What's the difference between the mean and expected value of a normal distribution?

My question might be a bit dumb but I'm confused so I'd like it if someone could clear this up for me. I've always thought that the mean of the normal distribution is equal to the expected value of ...
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Simple application of survival function of noncentral chi square distribution

Im looking for a new(ish) paper that is an application (with a simple setup and background) of the survival function of a noncentral (or with noncentrality parameter 0) chi square distribution. I want ...
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How does one define the sum of N random variables in Python? [closed]

Given $X_1 \cdots X_n \stackrel{iid}{\sim} exp(1)$ I want to show that $Y = 2\sum_{i=1}^{n}X_i \stackrel{}{\sim} \chi^2_{2n}$ I proved it by computing the MGF of Y as $M_{Y_1}(t) = M_{2\sum_{i=1}^{n}...
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Converting Log-Likelihood to Chi-square

I'm using two different algorithms to get a periodogram. One outputs log-likelihood and the other outputs chi-squared test statistic, but I would like a way to convert from log-likelihood to $\chi^2$ ...
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At chi squared qq plot, shouldn't normal samples be placed diagonally with chi-square?

I tried to draw a chi-square qq plot from sample following a bivariate normal distribution. This is my code: ...
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What does “having excellent Likelihoods” mean ? (MCMC code) [closed]

I asked an astrophysicist about MontePython code (MCMC code). He told me that its team had excellent Likelihoods about a cosmological survey. What does "having excellent Likelihoods" mean ? ...
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Non-centrality of likelihood ratio test statistic chi2 under alternate hypothesis

I am having trouble understanding how to determine the non-centrality parameter of the $\chi^2$ distribution symptotically followed by the likelihood ratio test statistic if the data follow the ...
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How to rearrange a ratio of Gamma functions to code it

I have to evaluate the following ratio: $\frac{\Gamma(\frac{x}{2} - \frac{1}{2})}{\Gamma(\frac{x}{2})}$ I am coding in MATLAB and I have this equation inside a loop. $x$ takes values between 2 and 400....
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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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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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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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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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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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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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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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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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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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143 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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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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