Question

(6) $\theta$  $\epsilon$   $\mathbb{R}$. We pick $n = 2m + 1 (m \geq 0)$ samples randomly from the population which distributes uniformly between the  interval of. $(\theta -\frac{1}{2} , \theta + \frac{1}{2})$ . Answer the following questions regarding the median of the samples $\widetilde{X_{n}}$

$\cdot$  Show that the distribution which $\widetilde{X_{n}}$ follows has the distribution as shown below.

$g_{n}(x) = \frac{(2m&plus;1)!}{m!m!}\left \{ \frac{1}{4} -(x-\theta )^2\right \}^m$     $(\theta -\frac{1}{2} < x < \theta &plus;\frac{1}{2})$

$\cdot$   Find the expected value of $\widetilde{X_{n}}$ .

$\cdot$  Show $Var[\widetilde{X_{n}}]$ = $\frac{1}{4(2m&plus;3)}$ .

$\cdot$ When $n \rightarrow \infty$ , show that $\widetilde{X_{n}}$ is the consistent estimator of $\theta$.

Answer 1

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