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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.

In probability theory and statistics, the chi-squared ($\chi^2$) distribution with $k$ degrees of freedom is the distribution of a sum of the squares of $k$ independent standard normal random variables. It is one of the most widely used probability distributions in inferential statistics (for example, in hypothesis testing or in construction of confidence intervals).

See:

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How is the $\chi^2_1$-distribution not a Gaussian?

The $\chi ^2_n$-distro is the distribution of the sum $Z_1 ^2+Z_2 ^2+...$ where the $Z_i$ are drown from a standard normal distribution. However, the standard normal distribution is a Gaussian bell curve, i.e.…
AlphaOmega
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Overlapping $\chi^2$ random variables

I have 3 independent random variables that follow $\chi^2$ laws, with $m$ and $n$ the degrees of freedom: \begin{align}A&\sim\chi^2_m\\B&\sim\chi^2_n\\C&\sim\chi^2_m\end{align} I am interested to know the conditional probability distribution of…
user340474
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What is an example of a chi-squared distribution?

I'm trying to figure out the usage of the $\chi^2$ distribution. In other words, in what kind of situations it occurs and how is it useful in that situations. I read the wikipedia definition of $\chi^2$. Can someone give an example of a $\chi^2$…
Can't Tell
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chi-square distribution with ${n-1}$ degree of freedom

Suppose that ${y_1}, ..., {y_n}$ is a random sample from an ${N(\mu,\sigma^2)}$ distribution. Then $$ {\sum_{i=1}^{n}{\frac {(y_{i}-\bar{y})^{2}}{\sigma^{2}}}}$$ has a $\chi^{2}_{n-1}$ distribution. Why is this the sum of ${\chi^{2}}$ distributions…
Sirius Hou
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Distribution for an infinite sum of weighted chi-squared distributions

Let $X_1,X_2,\ldots$ be an infinite list of independent normal variables $X_i\sim\mathcal N(0,1)$, for $i=0,1,\dots$. Consider the the sum $$Y=\sum_ir^iX_i^2,$$ where $r\in(0,1)$ is a parameter to weight (geometrically) each term in the sum. By…
user340474
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Understanding the relationship between the scaled inverse $\chi^2$ and inverse $\chi^2$ distributions

wikipedia says that Also, the scaled inverse chi-squared distribution is presented as the distribution for the inverse of the mean of ν squared deviates, rather than the inverse of their sum. The two distributions thus have the relation that…
DancingIceCream
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Distribution of differences of chi-squared statistics

I encountered the fact that differences in $\chi^2$-statistics again follow a $\chi^2$-statistic. I wondered why this is the case and how one could to show or even prove that?
Taufi
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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 $A_{1i}=n ^ {-1/2}$ for $1 \leq i \leq n$. $Z_1, ..., Z_n$ are…
user300652
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