Simulation<br /><ul><li>Simulation is a technique of manipulating a model of a system through a process of imitation.
In simulation, the performance of the system is simulated by artificially generating a large number of sampling experiments on the model of the system without observing the real system.</li></li></ul><li>Monte Carlo Methods<br /><ul><li>Thus in simulation a computation device is used or computations are done on paper.
The processes which are being simulated involve an element of chance they are referred to as Monte Carlo method.
The use of Monte Carlo simulation eliminates the cost of building and operating expensive equipments; it is used, for instance, in the study of collision of photons with electrons, the scattering of neutrons and similar complicated phenomena.</li></li></ul><li>Monte Carlo Methods<br /><ul><li>Monte Carlo methods are also useful in situations where direct experimentation is impossible, for example, in study of the spread of cholera epidemics which are not induced experimentally on human populations.
Monte Carlo techniques are sometimes applied to the solution of mathematical problems which actually cannot be solved by direct means or where a direct solution is too costly or requires too much time.</li></li></ul><li>Random Numbers<br /><ul><li>Although Monte Carlo methods are sometimes based on actual gambling devices, it is usually expedient to use tables of random digits or random numbers.
Tables of random numbers consist of many pages on which the digits of 0, 1, 2. … , and 9 are set down in a “random” fashion, much as they would appear if they were generated one at a time by a gambling device giving each digit an equal probability of being selected.</li></li></ul><li>Random Numbers<br />Tables of random numbers are constructed so that the digits can be regarded as values of a random variable having the discrete uniform distribution<br /> f(x) = 1/10 for x = 0, 1, 2, …, or 9.<br />They can be used to simulate values of any discrete random variable and even continuous random variables.<br />
Simulation <br />To avoid such waste of effort and time, we could have used the following scheme:<br />
Simulation (Continuous case)<br />To simulate the observation of continuous random variables we usually start with uniform random numbers and relate these to the distribution function of interest.<br />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<br />u = F(x) for x as x = F -1(u).<br />
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<br />xi = F -1(ui); i = 1, 2, …..,r.<br />
Uniform Random Numbers<br />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:<br /> Let u = .d1d2…….<br /> 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. <br />
Box-Mullar Method<br />Box-Mullar Method<br />Consider two independent standard normal random variables whose joint density is given by<br />
Box-Mullar Method<br />Under a change to polar coordinates, z1 = r cos, z2 = r sin, find the joint density of r and and further show that<br />(i) r and are independent and r and has uniform distribution on the interval from 0 to 2;<br />(ii) u1 = / 2 and u2 = 1 – have independent uniform distributions;<br />(iii) The following relations between (u1, u2) and (z1, z2) hold.<br />