L3: Discrete RVs Concept 1: Uniforms
S2: Real World Representations
We start with the understanding that perfect random numbers exist only in an imaginary world of perfection. All applications of this model to the real world will be filled with different kinds of flaws. We must consider HOW we plan to apply the probability model, in order to see if the flaws can be tolerated in the particular application. So we must consider how random numbers are used in the real world. Here is a list of some of the ways that random numbers are used:
1. To ensure fair and equitable distribution, lotteries are used to determine winners. In this case, it is important that the real world procedure used should give equal chances to all. An EXCEL random number generator, seeded by a randomly chosen time, would be good for this purpose. EXCEL produces pseudo-random numbers
2. Cryptography codes a message using a sequence of random numbers. Here it is very important that no one else should be able to replicate this sequence. The EXCEL random number generator would not be very useful for this purpose, since it is easy to replicate it. Other types of methods are useful in this context. Various types of physical processes produce numbers which are closer to "TRUE" random numbers, a better match for the ideal models
3. Simulations and Monte-Carlo Integration. Here the fact that random numbers are evenly spread out, and so provide a good spread over the range of possibilities is very useful. Again, a different property of randomness is useful here, and Low Discrepancy Sequences provide a well spread out set of numbers sometimes called quasi-random.
These concepts are too advanced for a beginning student. But the main point being made here is that it is a mistake to look for perfection in the real world. FIRST we determine WHY we want to use a probability model. THEN we look at how to create a suitable probability model for a real world process.