A UNIFORM Random variable has the property that it gives equal chances to all members of the population. The simplest uniform random variable has the property that it takes integer values 1,2,..,N and all of these N outcomes have equal probability 1/N. This is a THEORETICAL CONSTRUCT -- a model of an idealized situation. We will discuss later how this model can be replicated in reality.
Given a population with members labeled {1,2,...,N} make a simple random draw R. The serial number of the member drawn is a Uniform Random Variable U(R). Note that R is a random member of the population, and U(R) is one characteristic of this member, namely the serial number of this member.
Recall that this is a probability model of a hypothetical population, from which we are making an idealized random draw which is known to provide perfectly equal probabilities to all members. It is never possible to realize this ideal PERFECTLY in reality, but we can make good approximations which serve practical purposes.
CREATING A real world replica of the uniform random variable. Is there a way we can choose integers from among 1,2,...,N so that all of them have equal probabilities of being chosen? This is a major and difficult research problem, which has received a lot of attention from many statisticians. Creating a model with perfect theoretical properties is easy, since this is purely a mental construct. A real world illustration is difficult because all real world processes are complex and can rarely match the perfection of an ideal imaginary process. Several methods to produce approximately uniform random variables in the real world have been proposed, and all of them have advantages and disadvantages; none are perfect.
The outcome of efforts to create REAL WORLD replicas of the theoretical uniform random variables are random number generators built into computer programs. For example =RANDBETWEEN(1,N) is an EXCEL function which attempts to produce a random number which has equal probability of being any number between 1 and N. As discussed earlier, this is a PSEUDO RANDOM number -- it provides an approximation to the theoretical concept, which is defective in certain ways. For most practical purposes, these defects do not matter. But in some cases these defects might be very important.
Regarding Real World Replications of Uniform Random Numbers, an important issue is one of KNOWLEDGE: How do we know that all members have equal chances of being selected? This we can NEVER know for sure in the real world, because the real world is too complex to be analyzed by simplified models of any kind. In our hypothetical models, we do KNOW that all members have equal chances, because our model world is imaginary, perfect and idealized. It is a mental construct, and our IDEA that it is perfect guarantees that the model is perfect because our model is exactly the same as our IDEA of the model. This is because the model is a mental construct. The real world is complex and out there, and not a mental construct. It may or may not correspond to our ideas about it. We can never know for sure.