Suppose the random varaible Y has non-zero probability at 0,1,2,3,... (i.e. the support of Y is the set of non-negative integers).

Define a random variable W:

W=0 ,if Y=0,1,2,or 3

--=Y-3 ,if Y=4,5,...

Define a random variable Z:

Z=max{0,Y-3}=0 ,if Y≦3

--------------=Y-3 ,if Y>3

And I have 2 questions...

1) Can I say that W and Z areequal as random variables(i.e. W=Z) ?

(what is bothering me is that W is undefined at e.g. Y=0.5, Y=2.2, etc. while Z is defined everywhere, my notes say that W and Z are equal random varaibles, but I just struggle to understand why)

2) Is it true that E(W)=E(Z) ?

Hopefully someone can clarify this! Thank you!

Note: also under discussion in SOS math cyberboard