# Risk function

1. Mar 20, 2013

### autobot.d

1. The problem statement, all variables and given/known data
$X_{1} , ..., X_{5} \textit{ iid } N( \mu , 1) \textit{ and } \hat{\mu} = \bar{X}$
where
$L( \mu , \hat{\mu} ) = | \mu - \hat{\mu} |$

3. 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!

2. Mar 21, 2013

### Staff: Mentor

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

3. Mar 21, 2013

### 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.

4. Mar 21, 2013

### Ray Vickson

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?

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