# Questions tagged [chi-squared]

Use this tag for questions about (1) distributions of a sum of squares of independent standard normal random variables or (2) statistical hypothesis tests with such a sampling distribution if the null hypothesis is true.

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### Transform Wishart distribution to Chi-square distribution

This's actually what I'm trying to prove: $$\frac {a^{'}\Sigma^{-1}a}{a^{'}W^{-1}a} \sim \chi^{2}_{n-p+1}$$ $a$ is any P-dimensional nonzero constant vector, and $W \sim W_{p}(n,\Sigma)$, $\Sigma$ ...
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### Limit of expected value of reciprocal of $\chi^2_k$ for $k \to \infty$

I must figure out $\lim_{k \to \infty} \mathbb{E}\left[ W_k \right]$ for $$W_k = \frac{1}{a+b \cdot\chi^2_k} \qquad a,b \in \mathbb{R}_+$$ where $\chi^2_k$ indicates a chi-squared random ...
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### Distribution of sum of squared differences between two standard normal variables

The following question should be answered: Is the distribution of $\, T_n = (X_1 - X_2)^2 + ... + (X_{n-1}-X_n)^2$, (with $X_i \sim N(0,1) \,i.i.d.)$, $\chi^2$-distributed? For clarity, I made the ...
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### Chi-square issue

Chi-sq is supposed to be calculated $\sum_{i=1}^R \sum_{j=1}^C \frac{(O_{i,j} - e_{i,j})^2}{e_{i,j}}$. I thought there are two groups and it's even consistent with their chi-square estimate of which ...
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### Let X(X bar) denote the mean of a random sample of size 36 for Chi-squared distribution with 2 degrees of freedom.Find P(1.75 <X<2.5)

My strategy was to prove that the given chi squared distribution converges to a normal distribution and then approximate the probability as a Normal distribution, not sure if that's the correct way to ...
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### How to detect outliers with Gaussian mixture models?

Assuming a linear time-invariant system (LTI): $$x_t=Ax_{t-1}+Bu_t$$ $$y_t=Cx_t+w_t$$ where $x_t$ is the system output at timestamp $t$, $u_t$ is the input to the system at timestamp $t$, $y_t$ is the ...
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### How to obtain critical value from Chi-square pdf

I'm trying to learn about the Chi-square goodness of fit test for an assignment I'm doing, so far I've understood everything but critical values, I want to know where they come from. Searching online ...
Let $A \in \mathbb{R}^{N \times n}$ be a full row rank matrix and $b \in \mathbb{R}^{N}$ where all entries are drawn randomly, identically, and independently from $\mathcal{N}(0, 1)$. Let $f$ be the ...
### Why is Hotelling's $T^2 \sim \chi^2_p$ for large $n$?
I'm interested in some proof (simple if possible) as to why Hotelling's $T^2$ is chi-squared distributed for large n. I understand and can show that the Mahalanobis Distance is in fact chi-squared ...