Random sample in probability and statistics Is the word random sample has different definitions in probability and statistics?
In probability random sample is a vector of IID random variables.
I am confused with the same word in statistics. Does it contain the same definition in statistics? 
It seems to me like random sample in statistics is a subset of range of random sample in probability. Am i correct? 
If no, please give the exact definition in both streams.
 A: I'm not sure if I am familiar with "exact definitions," but I can describe the connection between simple random sampling (statistics) and a collection of i.i.d. random variables (probability). You might find it irrelevant to your question, in which case I'll delete my rambling and reflect on what shambles my life has become.

Simple random sampling
Given the human population of the world, a simple random sample of $1000$ people would be a randomly chosen subset of $1000$ people, such that each possible subset is equiprobable.
You can achieve this by choosing the first person uniformly at random from the entire population, then choosing the second person uniformly from the remaining people, and so on. This is sampling without replacement.
You could instead consider choosing the first person uniformly at random from the population, and then choosing the second person also uniformly at random from the entire population, and so on. In this way, the sample you get might have a person appear multiple times. This is sampling with replacement. When the population is large, sampling with replacement is roughly the same as sampling without replacement.

i.i.d. random variables
What does this have to do with i.i.d. random variables? Well, if you consider the sampling with replacement procedure I mentioned above, each person in the sample can be viewed as a uniform random variable over the human population. In particular, the sample of $1000$ people are simply $1000$ i.i.d. random variables.
More generally, instead of the uniform distribution over all humans, you can consider i.i.d. random variables drawn from some fixed arbitrary distribution, which is this more abstract notion of "random sample" you may find in probability and mathematical statistics. As I tried to explain above, it is very much related to and motivated by the more concrete notion of a sample in survey methodology.
