Many of the distributions discussed in this chapter are related to one another in various ways. For example, the geometric distribution is related to the binomial distribution. The geometric distribution represents the number of trials until the next success while the binomial represents the number of successes in a fixed number of trials. Similarly, the Poisson distribution is related to the exponential distribution. The exponential distribution represents the amount of time until the next occurrence of an event while the Poisson distribution represents the number of times an event occurs within a given period of time.
In some situations, as when the number of trials for the binomial distribution becomes very large, the normal and binomial distributions become similar. For these two distributions, as the number of binomial trials approaches infinity, the probabilities become identical for any given interval. For this reason, you can use the normal distribution to approximate the binomial distribution when the number of trials becomes too large for Crystal Ball to handle (more than 1000 trials). You also can use the Poisson distribution to approximate the binomial distribution when the number of trials is large, but there is little advantage to this since Crystal Ball takes a comparable amount of time to compute both distributions.
Likewise, the normal and Student’s t distributions are related. With Degrees of Freedom, then 30, Student’s t closely approximates the normal distribution.
The binomial and hypergeometric distributions are also closely related. As the number of trials and the population size increase, the hypergeometric trials tend to become independent like the binomial trials: the outcome of a single trial has a negligible effect on the probabilities of successive observations. The differences between these two types of distributions become important only when you are analyzing samples from relatively small populations. As with the Poisson and binomial distributions, Crystal Ball requires a similar amount of time to compute both the binomial and hypergeometric distributions.
The yes-no distribution is simply the binomial distribution with Trials = 1.
The Weibull distribution is flexible. Actually, it consists of a family of distributions that can assume the properties of several distributions. When the Weibull shape parameter is 1.0, the Weibull distribution is identical to the exponential distribution. The Weibull location parameter lets you set up an exponential distribution to start at a location other than 0.0. When the shape parameter is less than 1.0, the Weibull distribution becomes a steeply declining curve. A manufacturer might find this effect useful in describing part failures during a burn-in period. When the shape parameter is equal to 2.0, a special form of the Weibull distribution, called the Rayleigh distribution, results. A researcher might find the Rayleigh distribution useful for analyzing noise problems in communication systems or for use in reliability studies. When the shape parameter is set to 3.25, the Weibull distribution approximates the shape of the normal distribution; however, for applications when the normal distribution is appropriate, us it instead of the Weibull distribution.
The gamma distribution is also a flexible family of distributions. When the shape parameter is 1.0, the gamma distribution is identical to the exponential distribution. When the shape parameter is an integer greater than one, a special form of the gamma distribution, called the Erlang distribution, results. The Erlang distribution is especially useful in the areas of inventory control and queueing theory, where events tend to follow Poisson processes. Finally, when the shape parameter is an integer plus one half (e.g., 1.5, 2.5, etc.), the result is a chi-square distribution, useful for modeling the effects between the observed and expected outcomes of a random sampling.
When no other distribution seems to fit the historical data or accurately describes an uncertain variable, you can use the custom distribution to simulate almost any distribution. The Load Data button on the Custom Distribution dialog lets you read a series of data points or ranges from value cells in the worksheet. If you like, you can use the mouse to individually alter the probabilities and shapes of the data points and ranges so that they more accurately reflect the uncertain variable.