Okay, I accept that I'm wrong and you're right. Now the interesting part is that my mathematical intuition is not that great, but this is a pretty big fail even for it. So in between googling for crow recipes, I think I need to poke around my own mind and figure out which wrong turn did it happily take... I suspect I got confused about the expectation operator, but to confirm I'll need to drag my math intuition into the interrogation room and start asking it pointed questions.