4 Applications
So how do we apply models of uniform random variables in the real world? It is only context of real world applications that such models can be evaluated. One of the most important applications is for obtaining Simple Random Samples. Given a Real World Population (as opposed to a hypothetical population), how can we generate the analog of a simple random sample? Remember that GENUINE simple random samples exist only in an imaginary and ideal world. In the real world, we create imperfect replications, which have similar properties. This is just like true Euclidean Lines exist only in the Platonic world of ideals, but we can draw real world lines which have similar properties.
Problem: Give a population of 650 Students, generate a simple random sample of 50 students.
Solution: List the names of the students in numerical sequences from 1 to 650. Next, draw a sequence of uniform random integers which have possible outcomes in the set 1,2,...,650. This can be done in EXCEL by setting U1 = RANDBETWEEN(1,650); this generates the first uniform random number. Choose the student with the sequence number U1 for the first element of your random sample. Similarly set U2=RANDBETWEEN(1,650) as the second sequence number and continue this process to get to U50. That will generate a simple random sample of 50 students. .
Purpose -- generally speaking, we draw simple random samples when we want to study a small population and then make inferences about the big population. By the Law of Large Numbers, proportions of characteristics in the Simple Random Sample should be close to the proportion in the full population. We need to ensure that THIS PROPERTY of random draws holds for our Real World Random Number Generator. As long as the proportions in the true population are approximately matched to the proportions in the random sample, our statistical inference procedures will work properly. OTHER properties of random numbers need not be tested. This property has been tested for RANDBETWEEN and it works fine, so RANDBETWEEN is a suitable random generator for creating simple random samples. It is not a suitable random number generator for applications in cryptography, where high levels of security are desired.