Statistical Analysis Stack Exchange
Statistical Analysis Stack Exchange

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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Why is the sampling distribution of variance a chi-squared distribution?

The statement The sampling distribution of the sample variance is a chi-squared distribution with degree of freedom equals to $n-1$, where $n$ is the sample size (given that the random variable of interest is normally distributed). Source My…
Remi.b
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sum of noncentral Chi-square random variables

I need to find the distribution of the random variable $$Y=\sum_{i=1}^{n}(X_i)^2$$ where $X_i\sim{\cal{N}}(\mu_i,\sigma^2_i)$ and all $X_i$s are independent. I know that it is possible to first find the product of all moment generating functions…
pitfall
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Stouffer's Z-score method: what if we sum $z^2$ instead of $z$?

I am performing $N$ independent statistical tests with the same null hypothesis, and would like to combine the results into one $p$-value. It seems that there are two "accepted" methods: Fisher's method and Stouffer's method. My question is about…
amoeba
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Distribution of inverse Wishart to a power?

In a related question, I had asked about the norm induced by an inverse Wishart matrix. I am interested in generalizing that result somewhat. Let $A\sim\mathcal{W}_p\left(I,n\right)$, a Wishart matrix with scale matrix $I$, the identity, and $n$…
shabbychef
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Why is chi square used when creating a confidence interval for the variance?

This is a very basic question. Why do we use a chi square distribution? What is the meaning of this distribution? Why is this the distribution used for creating a confidence interval for the variance? Every place I google for an explanation just…
nafrtiti
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What is the sum of squared t variates?

Let $t_i$ be drawn i.i.d from a Student t distribution with $n$ degrees of freedom, for moderately sized $n$ (say less than 100). Define $$T = \sum_{1\le i \le k} t_i^2$$ Is $T$ distributed nearly as a chi-square with $k$ degrees of freedom? Is…
shabbychef
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pdf of the product of two independent random variables, normal and chi-square

what is the pdf of the product of two independent random variables X and Y, if X and Y are independent? X is normal distributed and Y is chi-square distributed. Z = XY if $X$ has normal distribution $$X\sim…
robin
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Expected value vs. most probable value (mode)

The expected value of a distribution $f(x)$ is the mean, that is the weighted average value $$E[x]=\int_{-\infty}^{+\infty} x \, \, f(x) dx$$ The most likely value is the mode, that is the most probable value. However do we expect somehow to see…
Sørën
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Relationship between gamma and chi-squared distribution

If $$Y=\sum_{i=1}^{N}X_i^2$$ where $X_i \sim \mathcal{N}(0,\sigma^2)$, i.e. all $X_i$ are i.i.d. normal random variables of zero mean with same variances, then $$Y \sim \Gamma\left(\frac{N}{2},2\sigma^2\right).$$ I know the chi-squared distribution…
kaka
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Distribution of the convolution of squared normal and chi-squared variables?

the following problem came up recently while analyzing data. If the random variable X follows a normal distribution and Y follows a $\chi^2_n$ distribution (with n dof), how is $Z = X^2 + Y^2$ distributed? Up to now I came up with the pdf of…
Leo Szilard
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Distribution of the ratio of dependent chi-square random variables

Assume that $ X = X_1 + X_2+\cdots+ X_n $ where $X_i \sim N(0,\sigma^2)$ are independent. My question is, what distribution does $$ Z = \frac{X^2}{X_1^2 + X_2^2 + \cdots + X_n^2}$$ follow? I know from here that the ratio of two chi-squared random…
x0dros
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Why does independence test use the chi-squared distribution?

The $\chi^2$ goodness-of-fit test uses the following statistic: $$ \chi_0^2=\sum_{i=1}^n\frac{(O_i-E_i)^2}{E_i} $$ In the test, granting that the conditions are met, one uses the $\chi^2$-distribution to calculate the p-value that given the $H_0$ is…
VF1
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What are the connections between: the normal, the $\chi^2$ & the F distributions?

I have often read that there is a huge connection between the normal distribution and several other distributions. But these were only mathematical explanations. What's the "real" connection between these 3 distributions (normal, $\chi^2$, and F)?
Le Max
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Why does the McNemar's test use $\chi^{2}$ and not the normal distribution?

I just noticed how the non exact McNemar's test uses the $\chi^{2}$ asymptotic distribution. But since the exact test (for the two case table) relies on the binomial distribution, how come it is not common to suggest the normal approximation to the…
Tal Galili
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Order statistics (e.g., minimum) of infinite collection of chi-square variates?

This is my first time here, so please let me know if I can clarify my question in any way (incl. formatting, tags, etc.). (And hopefully I can edit later!) I tried to find references, and tried to solve myself using induction, but failed at…
David M Kaplan
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