a) Why is it desirable to conduct Monte Carlo simulations using as many replications of the experiment as possible?

b. Explain in detail how pseudo-random numbers are generated in Excel.

c. In which range of numbers do we expect the random numbers drawn from a standard normal distribution to fall? Explain why.

ANSWER

a)

The choice of the number of replications used in simulation studies appear to be made solely by the judgment of the researchers; this is surmised due to the many simulation studies that have been conducted without any justification provided for the number of replications used. Currently, however, no empirically-based recommendations for general guidelines regarding the required number of replications a researcher should use in order to achieve accurate results exist. The obtained results from a Monte Carlo study might be invalid if too few replications were used, whereas time and resources may have been wasted if more replications were used than were necessary. In addition, with the same amount of time and resources but fewer replications, more conditions could be investigated.

b)

• The Excel RAND and RANDBETWEEN functions generate pseudo-random numbers from the Uniform distribution, aka rectangular distribution, where there is equal probability for all values that a random variable can take on. A good example of the uniform distribution is tossing a single die. The outcome of the toss is six possible values (1, 2, 3, 4, 5, 6) and each of these values is equally likely to occur. For a more scientific explanation, please check out wolfram.com.
• There is no way to seed either Excel RAND or RANDBETWEEN function, which are rumored to be initialized from the computer's system time. Technically, a seed is the starting point for generating a sequence of random numbers. And every time an Excel random function is called, a new seed is used that returns a unique random sequence. In other words, when using the random number generator in Excel, you cannot get a repeatable sequence with the RAND or RANDBETWEEN function, nor with VBA, nor by any other means.

c)

A distribution of values that cluster around an average (referred to as the “mean”) is known as a “normal” distribution. It is also called the Gaussian distribution (named for mathematician Carl Friedrich Gauss) or, if you are French, the Laplacian distribution (named for Pierre-Simon Laplace). Both mathematicians were working concurrently in the early nineteenth century on defining such a distribution.

The numbers work out as follows: Given a population, 68% of the members of that population will have values in the range of one standard deviation from the mean, 95% within two standard deviations, and 99.7% within three standard deviations. Given a standard deviation of 5 pixels, only 0.3% of the monkey heights will be less than 235 pixels (three standard deviations below the mean of 250) or greater than 265 pixels (three standard deviations above the mean of 250)