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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68 views

Suspicious about 2 ways to compute a variance : weighted sum of $\chi^2$ distrbution gives the same results than Moschopoulos distribution

In an astrophysics context, I am using sherical harmonics from Legendre decomposition. I take the definition of $a_{lm}$ following a normal distribution with mean equal to zero and take also the ...
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1answer
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Properties of $\chi^2(1)$ multiplied by a real value "$a$"

Is it true that the $\chi^2$ distribution with $k=1$ (noted $\chi^2(1)$) multiplied by a real value $a$ is equal to $\chi^2(a)$ ? If not, is there a particular distribution for $a\cdot\chi^2(1)$? If ...
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How do I determine the distribution of an additional sample $x_{n+1}$ given sample mean, variance $\bar x, s^2$ of a normally distributed $x$?

Suppose I draw $n$ samples $x_1...x_n$ of a random variable $x$, which is normally distributed with an unknown mean $\mu$ and variance $\sigma^2$. From those samples, I compute a sample mean $\bar x$ ...
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What is the distribution of [constant X chi square]?

I know that (n-1)*S^2 / sigma^2 ~ chi_square(df=n-1). ※ S^2 = sigma(Xi-X_bar)/(n-1) However, if n and sigma are known, what is the distribution of S^2? As far as I think, S^2/sigma^2 = Z^2 (Z ~ N(0,1))...
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Interpretation of Chi square distribution with 1/2 degrees of freedom

How do I interpret the Chi square ditsribution with 1/2 degrees of freedom? I know that with for instance n degrees of freedom the interpretation is that this is the number of squared standard normal ...
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38 views

Equation 10.2 from The Elements of Statistical Learning. Median of a chi-squared distribution

I'm reading about AdaBoost in the The Elements of Statistical Learning and I don't understand the equation 10.2. Below is an excerpt from the book. The power of AdaBoost to dramatically increase the ...
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1answer
52 views

Show that noncentral $X_2$ is $\chi^2(r-r_1,\theta -\theta_1)$

As I read a book named 'Introduction to mathematical statistics' written by Hogg et al, I stuck with the below question. $X_1$ and $X_2$ be two independent random variables. And Let $X_1$and $Y=X_1+...
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194 views

2 approaches for Monte-Carlo : weighted sum of $\chi^2$ distribution and Moschopoulos distribution with Gamma distribution

If I take as definition of $a_{lm}$ following a normal distribution with mean equal to zero and $C_\ell=\langle a_{lm}^2 \rangle=\text{Var}(a_{lm})$, and if I have a sum of $\chi^2$, can I write the 2 ...
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Variance in variance-weighted variance estimate?

Apologies for the confusing title, but I couldn't resist. Much can and has been said about computing the unbiased variance using a sample of points, weighting by the variances of each point (for ...
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For Poisson GLMs, when does the residual deviance follow a chi square distribution?

According to Generalized Linear Models by McCullagh and Nelder (I am looking at the 2nd edition, 1999), the deviance function is defined as $$D(y; \hat{\pi}) = 2[l(\tilde{\pi}; y)- l(\hat{\pi}; y)]$$ ...
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confused about unbiasedness of sample mean

Recently, in a different thread, I was convinced by others ( after claiming that they were wrong ) that the sample mean is unbiased for the mean of its underlying distribution. But the case of the chi-...
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Properties of $\Gamma$ distribution function that allow to include or not terms depending on multipoles - Computing variance [duplicate]

In the context of harmonic spherical decomposition where $C_\ell$ is the variance of $a_{lm}$ for a given multipole $\ell$, i.e $C_\ell = \langle |a_{lm}^2|\rangle$, I need help about the $\chi^2$ ...
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103 views

Expectation and distribution of ratio of correlated Gamma/Chi-square random variables

This question is very similar to: Distribution of the ratio of dependent chi-square random variables But the big difference is what happens when we don't have standard normal variables. I want to ...
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SaddlePoint Approximation Score Statistic

My question will be very practical. Actually, I work in statistical genetics and I have 1000 Score statistics for which I want to compute the corresponding p-values. To be more specific, I suppose a ...
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Deriving the Chi Square distribution [duplicate]

Let $X_1,...,X_n$ be IID random variable such that $X_i \sim N(0,1)$ for all $i=1,...,n$. Derive the distribution (i.e. give the pdf/cdf) of $\sum_{i=1}^{n} X_i^2$. I know that from the definition of ...
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Distribution of the sample mean Poisson's occurrence rate

Simple question. Given a sample of 100 values, the $\hat\lambda$ is calculated (e.g. with poissfit in Matlab), and the confidence intervals are provided. What's the ...
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1answer
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Derivation of F-distribution from inverse Chi-square?

I am trying to derive F-distribution from Chi-square and inverse Chi-square. Somewhere in process I make a mistake and result slightly differs from the canonical form of Fisher-Snedecor F distribution....
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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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1answer
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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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1answer
80 views

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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1answer
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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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1answer
264 views

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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1answer
50 views

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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1answer
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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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156 views

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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1answer
64 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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1answer
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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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1answer
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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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45 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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39 views

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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1answer
31 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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30 views

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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55 views

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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48 views

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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