Is "an independent and identically distributed RV with a pmf of..." a contradiction?

Question

I have a question in an assignment that begins by saying "generate $n$ i.i.d. pseudorandom outcomes for the random variable $X$ with a pmf of $[1/21, 2/21, 3/21, 4/21, 5/21, 6/21]$. Isn't this statement a contradiction as the outcomes of an i.i.d. all have the same probability? Or am I misunderstanding the definition of an I.i.d.?

Answer 1

Yes, you have misunderstood. I think you are confusing IID with the (discrete) uniform distribution. Here are some definitions from Wikipedia.

1. In probability theory and statistics, a sequence or other collection of random variables is independent and identically distributed (i.i.d. or iid or IID) if each random variable has the same probability distribution as the others and all are mutually independent.
2. In probability theory and statistics, the discrete uniform distribution is a symmetric probability distribution whereby a finite number of values are equally likely to be observed; every one of n values has equal probability 1/n.

So when you say " the outcomes of an i.i.d. all have the same probability", that is not true. If $X_1, \dotsc, X_n$ are IID, then they are independent and follow the same distribution, call it $F$. The distribution $F$ is not necessarily a uniform distribution. If it is, then your statement would be true. However, in your given example, $F$ is not uniform. Although it is not uniform, the generated outcomes are still IID. Omitting the IID condition would suggest to me that they are dependent somehow. In a simple, brief exercise, the condition might be dropped and simply be implied, but that might be considered a little careless or unwarranted.