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Briefly explain Monte-Carlo simulation with suitable example.
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The Monte-Carlo technique is a special technique of simulation, which involves the selection of random observations within the simulation model. This technique is restricted for applications involving random numbers to solve deterministic and stochastic problems. The principle of this technique is replacement of actual statistical system by another system described by some assumed probability distribution, and then sampling from this theoretical population by means of random numbers.

Consider the following example:

If a circle of radius R is inscribed inside a square with side length 2R, then the area of the circle will be $πR^2$ and the area of the square will be $(2R)^2$. So the ratio of the area of the circle to the area of the square will be π/4.

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It also means that if we pick a random point (x, y), where both x and y are between -1 &1, the probability of that this random point lies inside the unit circle is given as the proportion between the area of the unit circle and the square.

$$P (x^2 + y^2\lt 1) = \dfrac{A_{circle}}{A_{square}} = π4$$

So, if you pick N points at random inside the square, approximately Nπ/4 of those points should fall inside the circle.

M (no. of points inside the circle) = $\dfrac{Nπ}{4}$

So, supposing we pick ‘N’ random points, out of which ‘M’ of those fall inside the circle, we can then calculate π by:

$$π = \dfrac{4M}{N}$$

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