Hello,

Yes they're equal. Because Y is almost surely equal to 0,1,2,3,4,... So you can consider that Y being 2.2 never happens, so if W=2.2 it's a negligible event that shouldn't be taken into account.

If you want a more formal explanation :

2 rv W and Z are equal if there exists such that and , where .

But if is such that doesn't take integer values, then , where , since the support for Y is the set of natural integers.

So by taking A=B in the proposition (1), we get that the 2 rv's are (almost surely) equal.

And yes, their expectation is equal. By definition of each of them.