Evaluating the Expectation for $\mu$ and $\hat{\mu}$

[autobot.d]

Homework Statement

$X_{1} , ..., X_{5} \textit{ iid } N( \mu , 1) \textit{ and } \hat{\mu} = \bar{X}$
where
$L( \mu , \hat{\mu} ) = | \mu - \hat{\mu} |$

The Attempt at a Solution

$E[ | \mu - \hat{\mu} | ] = 0$
since
$E(\hat{\mu}) = \mu$

Am I missing something? Seems too easy.
Should I be using Indicator functions to handle the absolute values?
Thanks for the help!

 

[mfb]

$E(\mu - \hat{\mu})=0$ does not imply $E(|\mu - \hat{\mu}|)=0$.

 

[autobot.d]

Is there a reference you could point me to or a reason why?

Would it be true if
$\mu > \hat{\mu}$

and for

$\mu < \hat{\mu}$
$E[|\mu - \hat{\mu}|] = \frac{2}{\sqrt{\pi}}$


by going through and doing the actual integration.

 

[Ray Vickson]

Is there a reference you could point me to or a reason why?

Would it be true if
$\mu > \hat{\mu}$

and for

$\mu < \hat{\mu}$
$E[|\mu - \hat{\mu}|] = \frac{2}{\sqrt{\pi}}$


by going through and doing the actual integration.

You don't need a reference; you just need to stop and think for a moment. What kind of random variable Y could have E|Y| = 0?