Simulation involves imitating the performance of a system using a model to generate sample experiments without observing the real system. Monte Carlo simulation specifically involves an element of chance and is useful when direct experimentation is impossible or too costly. It generates random numbers that can represent values of random variables to simulate observations. Common techniques include using uniform random numbers between 0 and 1 to represent distributions and the Box-Muller method for generating normal random variables.

1. 3.2 Simulation

4. The processes which are being simulated involve an element of chance they are referred to as Monte Carlo method.

8. Simulation

9. Simulation To avoid such waste of effort and time, we could have used the following scheme:

10. Simulation (Continuous case) To simulate the observation of continuous random variables we usually start with uniform random numbers and relate these to the distribution function of interest. Let X is a continuous random variable with cumulative distribution function F(x), then U = F(X) is uniformly distributed on [0, 1]. So to find a random observation x of X, we select u an n-digit uniform random number and solve equation u = F(x) for x as x = F -1(u).

11. Further, to generate a random sample of size r from X, we take a sequence of r independent n-digit uniform random numbers say u1, u2, …., ur, and then generate x1, x2, …., xrwhere xi = F -1(ui); i = 1, 2, …..,r.

12. Uniform Random Numbers Uniform random numbers:A uniform random number u is a random observation from the uniform distribution on [0,1]. This can be done as under: Let u = .d1d2……. where the digits d1, d2, …… are independent and each diis chosen giving equal chance to the 10 digits 0, 1, 2, …, 9. We call u a uniform random number.

13. Box-Mullar Method Box-Mullar Method Consider two independent standard normal random variables whose joint density is given by

14. Box-Mullar Method Under a change to polar coordinates, z1 = r cos, z2 = r sin, find the joint density of r and  and further show that (i) r and  are independent and r and  has uniform distribution on the interval from 0 to 2; (ii) u1 =  / 2  and u2 = 1 – have independent uniform distributions; (iii) The following relations between (u1, u2) and (z1, z2) hold.