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?

  • $\begingroup$ Identically distributed suggests the same CDF. That would imply the same mean and variance (if these exist) but not the reverse. $\endgroup$ – Henry Jan 13 at 15:28
  • $\begingroup$ Random sampling (with replacement if necessary) may lead to identical distributions $\endgroup$ – Henry Jan 13 at 15:29
  • $\begingroup$ @Henry thank you for the clarification. Just a short follow-up re comment 1 to make sure I got this right. When you noted "but not the reserve" did it mean that normal variables with the same mean and variance do not necessarily have the same CDF? $\endgroup$ – PsychometStats Jan 13 at 15:37
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    $\begingroup$ Standard normal distributed random variables have identical distributions (Gaussian with mean 0 and variance 1). My point was there are other distributions with mean 0 and variance 1 which are not Gaussian (e.g. $\pm1$ with equal probabilities) and so not identically distributed to standard normals $\endgroup$ – Henry Jan 13 at 15:46
  • $\begingroup$ @Henry thank you so much, I got it!!! $\endgroup$ – PsychometStats Jan 13 at 15:48

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