A Question about an unbiased estimator

[Artusartos]

Homework Statement

The random sample $X_1, ... , X_n$ has a $N(0, \theta)$ distribution. So now I have to solve for c such that $Y= c \sum^n_{i=1}$ is an unbiased estimator for $\sqrt{\theta}$.

Homework Equations

The Attempt at a Solution

$E(c \sum^n_{i=1} |X_i|) = c \sum^n_{i=1} E(|X_i|) = c \sum^n_{i=1} \int \frac{|X_i|}{\sqrt{2(\pi)(\theta)}}e^{-X_i/(2\theta)}$

So now I have to solve...

$c \sum^n_{i=1} \int \frac{|X_i|}{\sqrt{2(\pi)(\theta)}} e^{-X_i/(2\theta)} = \sqrt(\theta)$, right? But how can I integrate the absolute value of $X_i$?

Thanks in advance

 

[haruspex]

Split the range of integration into x < 0, x > 0.

The random sample X1,...,Xn has a N(0,θ) distribution.

Do you mean, X1,...,Xn are independent samples from a N(0,θ) distribution? If so, why the subscript on θ_i?

 

[Artusartos]

Split the range of integration into x < 0, x > 0.

Do you mean, X1,...,Xn are independent samples from a N(0,θ) distribution? If so, why the subscript on θ_i?

Oh sorry, it's supposed to be just $\theta$