Monte Carlo Methods
Monte Carlo methods provide the researcher with estimated solutions that address a variety of mathematical problems by performing certain statistical sampling experiments.
Monte Carlo methods are the collection of different types of methods that perform the same process. The processes performed involve simulations using the method of random numbers and the theory of probability in order to obtain an approximate answer to the problem.
The defining characteristics of Monte Carlo methods involve the usage of random numbers in its simulations.
The researcher should note that Monte Carlo methods merely provide the researcher with an approximate answer. Thus, in the analysis involving Monte Carlo methods, the approximation of the error is a major factor that the researcher takes into account while evaluating the answers obtained from Monte Carlo methods.
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Types of Methods
The different types of Monte Carlo methods have different levels of accuracy, which also depends upon the nature of the question or problem which is to be addressed by the researcher. One of the vital uses of Monte Carlo methods involves the evaluation of the difficult integrals. Monte Carlo methods are applied especially in the cases where multi dimensional integrals are involved. Monte Carlo methods are valuable tools in cases when reasonable approximation is required in the case of multi dimensional integrals.
One of the Monte Carlo methods is a crude Monte Carlo method. This type of Monte Carlo method is used to solve the integral of a particular function, for example, f(x) under the limits ‘a’ and ‘b.’ In this type of Monte Carlo method, the researcher takes a number ‘N’ of the random sample, s. In this type of Monte Carlo method, the range on which the function is being integrated (i.e. a and b) is not equal the value of the sample size. The researcher in this type of Monte Carlo method finds the function value f(s) for the function f(x) in each random sample s. In this type of Monte Carlo method, the researcher then performs the summation of all these values and divides the result by ‘N’ in order to obtain the mean values from the sample. The researcher then performs the multiplication of that value by the integral (b-a) in order to obtain the integral.
Another type of Monte Carlo method is that of acceptance rejection Monte Carlo method. This type of Monte Carlo method is a flexible technique and is simple to understand. On the other hand, this type of Monte Carlo method gives one of the least approximate results among the four Monte Carlo methods. This method is helpful for the researcher to obtain the variance by adding up the variances for each sub interval.
One should use Monte Carlo methods because Monte Carlo methods can help solve complex problems. Additionally, Monte Carlo methods can approximate the answers very quickly which is otherwise very time consuming when the researcher is trying to determine an exact answer to the problem.
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