There can be many reasons why an e-mail is never sent, or sent but never received. I always make sure, if I can, that the sender e-mailaddres has the same domain name as the website, and that it is an existing e-mail address, most times this is enough to satisfy even very strict emailproviders. So setting the admin email is definitely a good idea. Something like noreply@xxxx.
But there are other considerations as well and other people might explain things far better then I evr could... Example: for one of my testservers only one domain name is acepted by the smtp server I use, its admin person being very very cautious... :)