Hello. Could I call your attention to the difference between
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{\displaystyle E({\frac {X}{n}})=E[E({\frac {X}{n}}|\theta )]=E(\theta )=\mu }
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{\displaystyle \mathrm {var} ({\frac {X}{n}})=E[\mathrm {var} ({\frac {X}{n}}|\theta )]+\mathrm {var} [E({\frac {X}{n}}|\theta )]}
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{\displaystyle =E[({\frac {1}{n}})\theta (1-\theta )|\mu ,M]+\mathrm {var} (\theta |\mu ,M)}
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{\displaystyle ={\frac {1}{n}}(\mu (1-\mu ))+{\frac {n_{i}-1}{n_{i}}}{\frac {(\mu (1-\mu ))}{M+1}}}
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{\displaystyle ={\frac {\mu (1-\mu )}{n}}(1+{\frac {n-1}{M+1}}).}
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{\displaystyle E\left({\frac {X}{n}}\right)=E\left[E\left({\frac {X}{n}}|\theta \right)\right]=E(\theta )=\mu }
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{\displaystyle \mathrm {var} \left({\frac {X}{n}}\right)=E\left[\mathrm {var} \left({\frac {X}{n}}|\theta \right)\right]+\mathrm {var} \left[E\left({\frac {X}{n}}|\theta \right)\right]}
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{\displaystyle =E\left[\left({\frac {1}{n}}\right)\theta (1-\theta )|\mu ,M\right]+\mathrm {var} \left(\theta |\mu ,M\right)}
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{\displaystyle ={\frac {1}{n}}\left(\mu (1-\mu )\right)+{\frac {n_{i}-1}{n_{i}}}{\frac {(\mu (1-\mu ))}{M+1}}}
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{\displaystyle ={\frac {\mu (1-\mu )}{n}}\left(1+{\frac {n-1}{M+1}}\right).}
I changed the former to the latter in Beta-binomial model . Michael Hardy 20:14, 19 October 2006 (UTC) Reply
Thanks
Charlesmartin14 22:32, 19 October 2006 (UTC) Reply
Charlesmartin14 22:43, 19 October 2006 (UTC) Reply
![{\displaystyle E({\frac {X}{n}})=E[E({\frac {X}{n}}|\theta )]=E(\theta )=\mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/1515c55ffb85b873b52245bcd6f30caef241faa9)
![{\displaystyle \mathrm {var} ({\frac {X}{n}})=E[\mathrm {var} ({\frac {X}{n}}|\theta )]+\mathrm {var} [E({\frac {X}{n}}|\theta )]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/27d4265582f8d6fd43ada444e4d896fa2fef8a2d)
![{\displaystyle =E[({\frac {1}{n}})\theta (1-\theta )|\mu ,M]+\mathrm {var} (\theta |\mu ,M)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/679e56b31a60fe1c4cef9f6700c24e4e8abc36ef)
![{\displaystyle E\left({\frac {X}{n}}\right)=E\left[E\left({\frac {X}{n}}|\theta \right)\right]=E(\theta )=\mu }](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe8a45c6c6c6fe1f222e05174c85eac787f0f2ea)
![{\displaystyle \mathrm {var} \left({\frac {X}{n}}\right)=E\left[\mathrm {var} \left({\frac {X}{n}}|\theta \right)\right]+\mathrm {var} \left[E\left({\frac {X}{n}}|\theta \right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7787bfdf6733a0354641bbdf67fa298c6f5f13aa)
![{\displaystyle =E\left[\left({\frac {1}{n}}\right)\theta (1-\theta )|\mu ,M\right]+\mathrm {var} \left(\theta |\mu ,M\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/49b2eb8babc6ed579334bca5f48d9460196815ee)
