Expectation for a $chi^2_n$ distributed random variable.












1














Consider $X_1,ldots, X_nsimmathcal{N}(mu,sigma^2)$ iid, where $mu$, $sigma^2$ are unknown and $c>0$. Define



$$S_n^2:=sum_{k=1}^nBig(X_k-frac{1}{n}sum_{i=1}^nX_iBig)^2$$



Now I want to calculate



$$E[cS_n^2-sigma^2]$$
Actually I was reading that $S_n^2$ might be $chi_n^2$-distributed, but I do not know why. So does someone has a hint on this problem?



Thanks in advance!










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  • Any source explaining the Bonferroni correction will prove $ES_n^2=(n-1)sigma^2$, which is enough for your purposes.
    – J.G.
    Nov 30 at 23:38










  • I was able to show it by simple calculation. Thanks for your comment!
    – user408858
    Dec 1 at 12:08
















1














Consider $X_1,ldots, X_nsimmathcal{N}(mu,sigma^2)$ iid, where $mu$, $sigma^2$ are unknown and $c>0$. Define



$$S_n^2:=sum_{k=1}^nBig(X_k-frac{1}{n}sum_{i=1}^nX_iBig)^2$$



Now I want to calculate



$$E[cS_n^2-sigma^2]$$
Actually I was reading that $S_n^2$ might be $chi_n^2$-distributed, but I do not know why. So does someone has a hint on this problem?



Thanks in advance!










share|cite|improve this question
























  • Any source explaining the Bonferroni correction will prove $ES_n^2=(n-1)sigma^2$, which is enough for your purposes.
    – J.G.
    Nov 30 at 23:38










  • I was able to show it by simple calculation. Thanks for your comment!
    – user408858
    Dec 1 at 12:08














1












1








1







Consider $X_1,ldots, X_nsimmathcal{N}(mu,sigma^2)$ iid, where $mu$, $sigma^2$ are unknown and $c>0$. Define



$$S_n^2:=sum_{k=1}^nBig(X_k-frac{1}{n}sum_{i=1}^nX_iBig)^2$$



Now I want to calculate



$$E[cS_n^2-sigma^2]$$
Actually I was reading that $S_n^2$ might be $chi_n^2$-distributed, but I do not know why. So does someone has a hint on this problem?



Thanks in advance!










share|cite|improve this question















Consider $X_1,ldots, X_nsimmathcal{N}(mu,sigma^2)$ iid, where $mu$, $sigma^2$ are unknown and $c>0$. Define



$$S_n^2:=sum_{k=1}^nBig(X_k-frac{1}{n}sum_{i=1}^nX_iBig)^2$$



Now I want to calculate



$$E[cS_n^2-sigma^2]$$
Actually I was reading that $S_n^2$ might be $chi_n^2$-distributed, but I do not know why. So does someone has a hint on this problem?



Thanks in advance!







probability-theory statistics stochastic-calculus stochastic-integrals descriptive-statistics






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share|cite|improve this question













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share|cite|improve this question








edited Nov 30 at 23:29

























asked Nov 30 at 17:28









user408858

344110




344110












  • Any source explaining the Bonferroni correction will prove $ES_n^2=(n-1)sigma^2$, which is enough for your purposes.
    – J.G.
    Nov 30 at 23:38










  • I was able to show it by simple calculation. Thanks for your comment!
    – user408858
    Dec 1 at 12:08


















  • Any source explaining the Bonferroni correction will prove $ES_n^2=(n-1)sigma^2$, which is enough for your purposes.
    – J.G.
    Nov 30 at 23:38










  • I was able to show it by simple calculation. Thanks for your comment!
    – user408858
    Dec 1 at 12:08
















Any source explaining the Bonferroni correction will prove $ES_n^2=(n-1)sigma^2$, which is enough for your purposes.
– J.G.
Nov 30 at 23:38




Any source explaining the Bonferroni correction will prove $ES_n^2=(n-1)sigma^2$, which is enough for your purposes.
– J.G.
Nov 30 at 23:38












I was able to show it by simple calculation. Thanks for your comment!
– user408858
Dec 1 at 12:08




I was able to show it by simple calculation. Thanks for your comment!
– user408858
Dec 1 at 12:08















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