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The Italian mathematician Gerolamo Cardano (1501–1576) stated without proof that the accuracies of empirical statistics tend to improve with the number of trials. This was then formalized as a law of large numbers. A special form of the LLN (for a binary random variable) was first proved by Jacob Bernoulli. It took him over 20 years to develop a sufficiently rigorous mathematical proof which was published in his Ars Conjectandi (The Art of Conjecturing) in 1713. He named this his "Golden Theorem" but it became generally known as "Bernoulli's theorem". This should not be confused with Bernoulli's principle, named after Jacob Bernoulli's nephew Daniel Bernoulli. In 1837, S. D. Poisson further described it under the name "la loi des grands nombres" ("the law of large numbers"). Thereafter, it was known under both names, but the "law of large numbers" is most frequently used.

After Bernoulli and Poisson published their efforts, other mathematicians also contributed to refinement of the law, including Chebyshev, Markov, Borel, Cantelli, Kolmogorov and Khinchin. Markov showed that the law can apply to a random variable that does not have a finite variance under some other weaker assumption, and Khinchin showed in 1929 that if the series consists of independent identically distributed random variables, it suffices that the expected value exists for the weak law of large numbers to be true. These further studies have given rise to two prominent forms of the LLN. One is called the "weak" law and the other the "strong" law, in reference to two different modes of convergence of the cumulative sample means to the expected value; in particular, as explained below, the strong form implies the weak.

Applications

One application of the Law of Large Numbers is the use of a important method of approximation, the Monte Carlo Method [citation]. This method uses a random sampling of numbers to approximate numerical results. The algorithm to compute an integral of f(x) on an interval [a,b] is as follows[citation]:


 * 1) Simulate uniform random variables X1, X2, ..., Xn which can be done using a software, and use a random number table that gives U1, U2, ..., Un  independent and identically distributed (i.i.d.) random variables on [0,1]. Then let Xi = a+(b - a)Ui for i= 1, 2, ..., n. Then X1, X2, ..., Xn are independent and identically distributed uniform random variables on [a, b].
 * 2) Evaluate f(X1), f(X2), ..., f(Xn)
 * 3) Take the average of f(X1), f(X2), ..., f(Xn) by computing (b-a)$$\tfrac{f(X1) + f(X2) +...+ f(Xn)}{n}$$ and then by the Strong Law of Large Numbers, this converges to (b-a)E(f(X1))=(b-a)$$\int_{a}^{b} f(x)\tfrac{1}{b-a}$$ =$$\int_{a}^{b} f(x)$$

We can find the integral of g(x) = cos2(x)$$\sqrt{x^3+1}$$ on [-1,2]. Using traditional methods to compute this integral is very difficult, so the Monte Carlo Method can be used here [citation]. Using the above algorithm, we get

$$\int_{-1}^{2} g(x)$$ = 0.905 when n=25

and

$$\int_{-1}^{2} g(x)$$ = 1.028 when n=250

We observe that as n increases, the numerical value also increases. When we get the actual results for the integral we get

$$\int_{-1}^{2} g(x)$$ = 1.000194

By using the Law of Large Numbers, the approximation of the integral was more accurate and was closer to its true value.