Independent and Identically Distributed and the Discrete Uniform Distribution

Question

I am learning about IID. I would like to know from expert if it is accurate to say that the Discrete Uniform Distribution is actually an example of IID random number generator? (if is accurate to say things that way?).

Second question. Wikipedia indicates that "A sequence of fair or loaded dice rolls is i.i.d.". I understand of IIDs that all random variables need to have the same probability. Thus if the dice roll is fair, yes I understand this is i.i.d however doesn't understand why if the dice roll is unfair this would be i.i.d too? (imagine for example that the number 2 has the probability to be drawn twice as more often than the other numbers, then clearly number 2 doesn't have the same probability than the other discrete numbers and this wouldn't be i.i.d?).

Thank you.

Answer 1

You can't talk about one variable being IID (independent and identically distributed) -- IID is a property of a sequence of random variables.

Further, it has nothing to do with uniformity, although there are plenty of ways to come up with sequences of IID variables that have uniform distributions.

Consider having a loaded coin: it lands on heads with probability $\frac{2}{3}$, and tails with probability $\frac{1}{3}$.

Say that we flip the coin twice. Does the outcome of the first coin flip change the outcome of the second? No! So, they are independent. Further, any given outcome (heads, tails) has the same probability for the first flip as it does for the second: heads is always $\frac{2}{3}$, and tails is always $\frac{1}{3}$. This is what we mean by identically distributed. Do you see how this is different than saying that each out come has the same probability (uniform)?

So, the outcomes of these two coin flips are IID random variables.