How do we derive that $S^2$ is chi-squared distributed (with $n-1$ df)?
Nov 12, 2017 · I am asking specifically how we can rigorously derive that it has the distribution that it has. EDIT 2: Those who have marked this question as a duplicate of this one may be …
Distribution of the sample variance - Cross Validated
When the observations are independent identically distributed with an unknown variance you have (n-1)S2 2 / σ σ 2 2 is a pivotal quantity allowing you to generate confidence intervals or test an …
probability - Variance of $S^2$ taken from Normal Distribution ...
Jan 30, 2018 · Explore related questions probability statistics probability-distributions normal-distribution sampling See similar questions with these tags.
Proof of $\\frac{(n-1)S^2}{\\sigma^2} \\sim \\chi^2_{n-1}$
May 6, 2016 · It must follow that b b (n − 1)S2/σ2 (n 1) S 2 / σ 2 has the same distribution as the sum of squares of n − 1 n 1 standard normals -- so by definition it has chi-squared (n − 1 n 1) distribution.
Variance of sample variance? - Mathematics Stack Exchange
Remember that (n − 1)S2/σ2 (n 1) S 2 / σ 2 is only guaranteed to be χ2 χ 2 when the sample is taken from a normal distribution, though.
Prove the sampling distribution of $S^2$ has the mean $\sigma^2
Nov 13, 2018 · You have the distribution of $\frac {n-1} {\sigma^2}S^2\sim\chi^2_ {n-1}$ by (2), so basically compute the expected value and variance of $\chi^2_ {n-1}$.
normal distribution - Why do we use $S^2$ while estimating the …
Oct 18, 2019 · Why do we use S2 S 2 while estimating the variance? Ask Question Asked 5 years, 7 months ago Modified 5 years, 7 months ago
Proof of the independence of the sample mean and sample variance
Then (n−1)S2 σ2 ∼χ2 n−1 (n 1) S 2 σ 2 ∼ χ n 1 2 so S2 S 2 is ancillary, while X¯ X is complete sufficient, and hence they are independent for all μ μ and our fixed σ2 σ 2. Since σ2 σ 2 was …
Probability density function (pdf) of normal sample variance ($S^2$)
It is for a theory class, but we have no homework. He mentioned it would be a good excerise to prove that (X-bar, S^2) are sufficient statistics for (mu, sigma^2) using the ratio method (not factorization …
How to prove $s^2$ is a consistent estimator of $\\sigma^2$?
Nov 14, 2020 · I am trying to prove that s2 = 1 n−1 ∑n i=1(Xi −X¯)2 s 2 = 1 n 1 ∑ i = 1 n (X i X) 2 is a consistent estimator of σ2 σ 2 (variance), meaning that as the sample size n n approaches ∞ ∞ , …