Mathematics, 11.06.2020 04:57 teresagonzalez6652
We are interested in finding an estimator for Var (Xi), and propose to use V=-n (1-Xn). 0/2 puntos (calificable) Now, we are interested in the bias of V. Compute: E [V]-Var (Xi)-[n Using this, find an unbiased estimator V for p (1 - p) if n22. rite barX_n for Л n . 72 1--X 7t
Answers: 2
and propose to use :
![E \ \ [ \bar V] - Var (X_i) =](/tpl/images/0682/6480/c9baa.png)
![[ \bar V]](/tpl/images/0682/6480/95d7d.png)
for 
![E( \bar X^2 _ n) = Var (\bar X_n) + [E(\bar X_n)]^2 \\ \\ = \dfrac{p(1-p)}{n}+ p \\ \\ = p^2 + \dfrac{p(1-p)}{n} \\ \\ \\ \hat V = \bar X_n (1- \bar X_n ) = \bar X_n - \bar X_n ^2 \\ \\ E [ \hat V] = E [ \bar X_n - \bar X_n^2] \\ \\ = E[\bar X_n ] - E [\bar X^2_n] \\ \\ = p-(p^2 + \dfrac{p(1-p)}{n}) \\ \\ = p-p^2 -\dfrac{p(1-p)}{n}](/tpl/images/0682/6480/44115.png)
![=p(1-p)[1-\dfrac{1}{n}] = p(1-p)\dfrac{n-1}{n}](/tpl/images/0682/6480/bf80c.png)
![Bias \ (\bar V ) = E ( \hat V) - Var (X_i) \\ \\ = p(1-p) [1-\dfrac{1}{n}] - p(1-p) \\ \\ - \dfrac{p(1-p)}{n}](/tpl/images/0682/6480/953ea.png)
![E [\hat V] = p(1-p ) \dfrac{n-1}{n}](/tpl/images/0682/6480/f35a4.png)
![E [\dfrac{n}{n-1} \ \ \bar V] = p(1 -p)](/tpl/images/0682/6480/653ff.png)
![E [\dfrac{n}{n-1} \ \ \bar X_n (1- \bar X_n )] = p (1-p)](/tpl/images/0682/6480/b6b42.png)
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We are interested in finding an estimator for Var (Xi), and propose to use V=-n (1-Xn). 0/2 puntos (...
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