Editing Law of large numbers (section)

== Applications ==
One application of the law of large numbers is an important method of approximation known as the [[Monte Carlo method]],<ref name=":1" /> which 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:<ref name=":1" />

# Simulate uniform random variables X<sub>1</sub>, X<sub>2</sub>, ..., X<sub>n</sub> which can be done using a software, and use a random number table that gives U<sub>1</sub>, U<sub>2</sub>, ..., U<sub>n</sub> independent and identically distributed (i.i.d.) random variables on [0,1]. Then let X<sub>i</sub> = a+(b - a)U<sub>i</sub> for i= 1, 2, ..., n. Then X<sub>1</sub>, X<sub>2</sub>, ..., X<sub>n</sub> are independent and identically distributed uniform random variables on [a, b].
# Evaluate f(X<sub>1</sub>), f(X<sub>2</sub>), ..., f(X<sub>n</sub>)
# Take the average of f(X<sub>1</sub>), f(X<sub>2</sub>), ..., f(X<sub>n</sub>) by computing <math>(b-a)\tfrac{f(X_1)+f(X_2)+...+f(X_n)}{n}</math> and then by the strong law of large numbers, this converges to <math>(b-a)E(f(X_1))</math> = <math>(b-a)\int_{a}^{b} f(x)\tfrac{1}{b-a}{dx}</math> =<math>\int_{a}^{b} f(x){dx}</math>

We can find the integral of <math>f(x) = cos^2(x)\sqrt{x^3+1}</math> on [-1,2]. Using traditional methods to compute this integral is very difficult, so the Monte Carlo method can be used here.<ref name=":1" /> Using the above algorithm, we get

<math>\int_{-1}^{2} f(x){dx}</math>  = 0.905 when n=25

and

<math>\int_{-1}^{2} f(x){dx}</math>  = 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

<math>\int_{-1}^{2} f(x){dx}</math> = 1.000194

When the LLN was used, the approximation of the integral was closer to its true value, and thus more accurate.<ref name=":1" />

Another example is the integration of <big>f(x) =</big> <math>\frac{e^x-1}{e-1}</math> on [0,1].<ref name=":2">{{Citation |last=Reiter |first=Detlev |title=The Monte Carlo Method, an Introduction |date=2008 |url=http://link.springer.com/10.1007/978-3-540-74686-7_3 |work=Computational Many-Particle Physics |series=Lecture Notes in Physics |volume=739 |pages=63–78 |editor-last=Fehske |editor-first=H. |access-date=2023-12-08 |place=Berlin, Heidelberg |publisher=Springer Berlin Heidelberg |language=en |doi=10.1007/978-3-540-74686-7_3 |isbn=978-3-540-74685-0 |editor2-last=Schneider |editor2-first=R. |editor3-last=Weiße |editor3-first=A.}}</ref> Using the Monte Carlo method and the LLN, we can see that as the number of samples increases, the numerical value gets closer to 0.4180233.<ref name=":2" />