# Approximation of one distribution with another

There are many situations where it is convenient or just plain necessary to approximate one distribution with another because of the numerical constraints involved. ModelRisk implements very precise approximations where it is necessary, so this topic is not essential reading to be able to build your risk analysis model. Nonetheless, understanding approximations helps get a better feel for how distributions relate to each other, and the difficulties sometimes faced in calculating probabilities of, or generating values from, discrete distributions.

For example, if we toss a coin one million times, how many heads will there be?  The appropriate distribution is Binomial(1 000 000, 0.5) but such a distribution is impractical to calculate. For a start, you would need to calculate every factorial for integers between 0 and a million. However, under certain conditions, the Binomial(n, p) distribution is very well approximated by a Normal(np, (np(1-p))Ѕ) distribution. In our example, that would mean using a Normal(500 000, 500) and we could readily calculate, for example, the probability of having exactly 501 000 heads by using the Normal probability density function:

or

=VoseNormalProb(501000,500000,500,0)

This section looks at a number of approximations, why they work and how to use them. It also provides you with a better understanding of the inter-relationships between the most common distributions.

Before proceeding, it is worthwhile reading about the Central Limit Theorem. because it explains the Normal distribution approximation to most of the distributions offered here.

We offer the following examples of approximations of distributions by other distributions:

The section on recursive distributions shows you how you can create precise discrete distributions where the parameters have become unmanageably large to deal with in the normal way.

In addition to approximating distributions by other distribution, it sometimes is useful to create your own distribution, which can be done in four ways, as explained here.