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?
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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: