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In probability theory and statistics, the negative binomial distribution is a discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of successes occurs. For example, we can define rolling a 6 on a dice as a success, and rolling any other number as a failure, and ask how many failure rolls will occur before we see the third success. In such a case, the probability distribution of the number of failures that appear will be a negative binomial distribution.
The Pareto distribution, named after the Italian civil engineer, economist, and sociologist Vilfredo Pareto, is a power-law probability distribution that is used in description of social, quality control, scientific, geophysical, actuarial, and many other types of observable phenomena; the principle originally applied to describing the distribution of wealth in a society, fitting the trend that a large portion of wealth is held by a small fraction of the population. The Pareto principle or "80-20 rule" stating that 80% of outcomes are due to 20% of causes was named in honour of Pareto, but the concepts are distinct, and only Pareto distributions with shape value of log45 ≈ 1.16 precisely reflect it. Empirical observation has shown that this 80-20 distribution fits a wide range of cases, including natural phenomena and human activities.
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In probability and statistics, the Gompertz distribution is a continuous probability distribution, named after Benjamin Gompertz. The Gompertz distribution is often applied to describe the distribution of adult lifespans by demographers and actuaries. Related fields of science such as biology and gerontology also considered the Gompertz distribution for the analysis of survival. More recently, computer scientists have also started to model the failure rates of computer code by the Gompertz distribution. In Marketing Science, it has been used as an individual-level simulation for customer lifetime value modeling. In network theory, particularly the Erdős–Rényi model, the walk length of a random self-avoiding walk (SAW) is distributed according to the Gompertz distribution.
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In probability theory, a beta negative binomial distribution is the probability distribution of a discrete random variable equal to the number of failures needed to get successes in a sequence of independent Bernoulli trials. The probability of success on each trial stays constant within any given experiment but varies across different experiments following a beta distribution. Thus the distribution is a compound probability distribution.
In probability and statistics, the Gamma/Gompertz distribution is a continuous probability distribution. It has been used as an aggregate-level model of customer lifetime and a model of mortality risks.
In probability theory and statistics, the discrete Weibull distribution is the discrete variant of the Weibull distribution. The Discrete Weibull Distribution, first introduced by Toshio Nakagawa and Shunji Osaki, is a discrete analog of the continuous Weibull distribution, predominantly used in reliability engineering. It is particularly applicable for modeling failure data measured in discrete units like cycles or shocks. This distribution provides a versatile tool for analyzing scenarios where the timing of events is counted in distinct intervals, making it distinctively useful in fields that deal with discrete data patterns and reliability analysis.
References
- ↑ McCarthy, Daniel. "Buy 'Til You Die - A Walkthrough" (PDF). Buy ’Til You Die - A Walkthrough. Retrieved 21 March 2019.
- 1 2 3 Fader; et al. (Spring 2005). ""Counting Your Customers" the Easy Way: An Alternative to the Pareto/NBD Model" (PDF). Marketing Science. 24 (2): 275–284. doi:10.1287/mksc.1040.0098 . Retrieved 21 March 2019.
- ↑ Fader; et al. (November–December 2010). "Customer-Base Analysis in a Discrete-Time Noncontractual Setting" (PDF). Marketing Science. 29 (6): 1086–1108. doi:10.1287/mksc.1100.0580 . Retrieved 21 March 2019.
- ↑ Schmittlein, David; Morrison, Donald; Colombo, Richard. "Counting Your Customers: Who-Are They and What Will They Do Next?". Management Science. 33 (1): 1–24. doi:10.1287/mnsc.33.1.1.