I'm using the R language to generate a sample of 32000 independent values Bernoulli (q) with q = 0.1 to construct a dot plot with the respective averages calculated using the first N values of the sample with N = 2, 3, ..., 32000. Is it a coincidence that the average for N max (32000) is very close to the value "q" of probability? have any statistical explanation?. closer to q. Do you have anything to do with expectation of a random variable that follows a Bernoulli-type distribution is equal to q?
is because under the law of large numbers states that if X1, X2, X3, ... is an infinite sequence of independent random variables that have the same expected value μ and variance σ2,
then the average converges in probability to μ. In other words, for any positive number ε we have: