Relationship between normal distribution and chi | Chi-square distribution normal distribution
Index:TheBookofStatisticalProofs[1]▷ProbabilityDistributions[2]▷Univariatecontinuousdistributions▷Normaldistribution▷Relationshiptochi-squareddistributionTheorem:Let$X_1,ldots,X_n$beindependent[3]randomvariables[4]whereeachofthemisfollowinganormaldistribution[5]withmean$mu$andvariance$sigma2$:[label{eq:norm}X_isimmathcal{N}(mu,sigma2)quadext{for}quadi=1,ldots,n;.]Definethesamplemean[6][label{eq:mean-samp}ar{X}=frac{1}{n}sum_{i=1}{n}X_i]andtheunbiasedsamplevariance[7][label{eq:var-samp}s2=fr...
Index: The Book of Statistical Proofs[1] ▷ Probability Distributions[2] ▷ Univariate continuous distributions ▷ Normal distribution ▷ Relationship to chi-squared distributionTheorem: Let $X_1, ldots, X_n$ be independent[3] random variables[4] where each of them is following a normal distribution[5] with mean $mu$ and variance $sigma2$:
[label{eq:norm} X_i sim mathcal{N}(mu, sigma2) quad ext{for} quad i = 1, ldots, n ; .]Define the sample mean[6]
[label{eq:mean-samp} ar{X} = frac{1}{n} sum_{i=1}{n} X_i]and the unbiased sample variance[7]
[label{eq:var-samp} s2 = frac{1}{n-1} sum_{i=1}{n} left( X_i - ar{X} ight)2 ; .]Then, the sampling distribution[8] of the sample variance is given by a chi-squared distribution[9] with $n-1$ degrees of freedom:
[label{eq:norm-chi2} V = (n-1) , frac{s2}{sigma2} sim chi2(n-1) ; .]Proof: Consider the random variable[10] $U_i$ defined as
[label{eq:Ui} U_i = ...