Part 1: The extended family of CB Assumptions

In the last post, we discussed the way to model two distributions: Erlang and two-parameter lognormal. In this post, we will continue with a few more distributions that can be simulated in CB by modifying one or more parameters of one of the existing distributions. Data can also be fitted to these distributions by locking one or more parameters to specific values. For a detailed discussion on locking parameters while fitting to distributions, check out our coverage of two-parameter lognormal distribution in the previous post, and our help documents.

**Example 3: The Maxwell distribution or Maxwell-Boltzman distribution**This distribution is used in statistical physics, specifically as the distribution of molecular speeds in thermal equilibrium.Mathematically, this is a variant of the gamma distribution.

Check out the Wikipedia entry and the MathWorld website.**Notes and formulas:**This distribution is a special form of the gamma distribution. A Maxwell-Boltzman distribution with parameter 'a' can be modeled by a gamma distribution with location = 0, scale = 2a**Quick summary:**^{2}and shape = 3/2.Define a gamma distribution assumption with location = 0, scale = 2a**Generate random numbers:**^{2}and shape = 3/2, where 'a' is the parameter of the distribution.To fit a dataset to this distribution, we have to lock both the location and shape of the gamma distribution to 0 and 3/2 respectively. The parameter 'a' can be found out from the fitted scale. If the fitted scale is 's', then: a = sqrt(s/2).**Fit to this distribution:**

**Example 4: The Chi-square distribution**This distribution is commonly used in statistical inference. One of the common Goodness-of-fit (GOF) statistic used in distribution fitting is the Chi-squared statistic, which, of course, follows the Chi-square distribution. To model this distribution in CB, we will use the same technique that we used to model the two-parameter lognormal distribution.

Check out the Wikipedia entry and the NIST website.**Notes and formulas:**This distribution is a special form of the gamma distribution. A Chi-square distribution with 'd' degrees of freedom can be modeled by a gamma distribution with location = 0, scale = 2 and shape = d/2. We use this method in CB to directly construct the specific Chi-squared distribution and calculate the critical values (p-values) of the Chi-squared GOF in distribution fitting, that is reported in the distribution fitting results window.**Quick summary:**Define a gamma distribution assumption with location = 0, scale = 2 and shape = d/2, where 'd' is the degrees of freedom of the Chi-square distribution.**Generate random numbers:**Fitting to a Chi-square distribution is slightly tricky, since we do not have the ability to lock the scale of a gamma distribution in distribution fitting. We will come back to this later in a future post in this series.**Fit to this distribution:**

**Example 5: The Rayleigh distribution**Often used in the physical sciences, this distribution is a special case of the Weibull distribution.

Check out the Wikipedia entry.**Notes and formulas:**This distribution is a special form of the weibull distribution with shape = 2.**Quick summary:**Define a Weibull distribution assumption with shape = 2.**Generate random numbers:**Fitting to a Rayleigh distribution is also easy, just lock the shape value of the Weibull distribution to 2 while fitting.**Fit to this distribution:**

**Example 6: The Pearson type V distribution or Inverse gamma distribution**Mostly used to measure the time taken to perform a task. Also known as inverse gamma distribution.

Check out the Wikipedia entry.**Notes and formulas:**This distribution can be modeled using gamma distribution. If X ~ PearsonV(scale = a, shape = b), then Y = 1/X ~ gamma(Location = 0, scale = 1/a, shape = b), so it follows that X = 1/Y.**Quick summary:**Given the parameters of the PearsonV distribution (scale=a, shape=b), set up a gamma distribution with parameters: Location = 0, scale = 1/a, shape= b. Next, set up a forecast having the inverse of the gamma assumption. As you run the simulation, the forecast values would represent the random numbers from Pearson-V distribution.**Generate random numbers:**

Figure 1: Excel worksheet for simulating Pearson-V distribution in CB |

Figure 2: CB chart showing the Pearson-V distribution with statistics |

Fitting to a Pearson-V distribution is also easy. Given a dataset, follow the steps below:**Fit to this distribution:**- Calculate the inverse of the values
- Fit these values to a gamma distribution with location locked to 0.
- The scale of the PearsonV is the inverse of the fitted gamma scale. The shape of both the distributions are the same.

Figure 3: Fitting data to a Pearson V distributio |