Usually, random variables are written in block capitals, that's why this definition is a bit confusing. In fact, are independent copies of a random variable , i.e. they are independent identically distributed random variables (and their common distribution is that of ). This is the way to model " values drawn from a random variable ", as you wrote. For a given sample , a random variable assumes only one value . When modeling several independent repetitions an experiment (like throwing a coin), the sample space has to be widened so that it contains the values of every sequence that can appear. Numerical observations would be for some in the sample space.

I hope this makes things clearer...

Laurent.