Q: Are random variables acting as a tool for us to be able to create multiple iterations (to measure and average) of an otherwise deterministic experiment?
A: An answer has two ingredients:
- firstly, many deterministic questions can be formulated as asking for the expectation of a random variable. For example, the integral $\int_a^b f(x)dx$ is the expectation of $(b-a)f(x)$ for a random variable $x$ which is uniformly distributed in the interval $(a,b)$.
- secondly, the law of large numbers allows us to approximate the expectation by random sampling.
This is the essence of the Monte Carlo method. A further refinement could then be to reduce the variance of the estimator (importance sampling).
Q: Follow-up question: Why doesn't each iteration simply add random noise to the data?
A: After $N$ iterations, the signal has increased by a factor $N$, while the uncertainty due to the noise has increased by a factor $\sqrt N$, hence the relative uncertainty, which is what matters for the expectation, decays as $1/\sqrt N$.