Let $Z_1, ..., Z_k$ be identically and independently distributed (i.i.d.) set of standard normal random variables.
I understand that as part of the i.i.d. independent broadly means that variables occur independently of one another, meaning they can be multiplied. But I would like to ask a couple of basic questions please.
Q1: Can somebody clarify what exactly is meant by identically distributed ? Does it simply mean that each $Z_k$ variable has the same distributional properties, i.e. the mean and variance; or does it mean something else?
Q2: I am aware that the i.i.d. is an assumption in itself, but in practice $Z_k$ variables rarely, if ever, have identical distributional properties. Is this fact usually acknowledged and ignored or as long as distributional properties of each variable are similar, it is deemed to be a satisfactory situation?