Explain
how the "Law of Large Numbers" combines with the "General Definition" of Expected Value which uses an integral with limits of
negative infinity to infinity.

The law of large numbers states that in a very large number of trials the percentage difference between observed value and the expected value becomes zero. The observed value may be less (negative) or more (positive) giving rise to a normal distribution. Also remember any normal distribution can be standardized to a N(0,1).