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?