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Author: Admin | 2025-04-28
These events occur with a known constant mean rate and independently of the time since the last event. The problem with the Poisson Distribution is that it is discrete and not continuous. The Poisson Distribution deals with the number of occurrences *in a fixed period of time. *This is not the way we want to look at the Luck Statistic.The next step is to check the Gamma Distribution, which is continuous. The Gamma Distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. The gamma distribution, predicts the time until the k-th event occurs. When the shape parameter of the Gamma Distribution has an integer value, the distribution is called the Erlang Distribution. This is important for looking at the Luck Statistic because it will always be a positive integer.The Luck Statistic is negative binomially distributed, so can be approximated using the Erlang Distribution.Erlang DistributionWe don’t need to go deep into the formula of this distribution, but the Erlang Distribution can be thought of as a generalization of the exponential distribution, a continuous distribution that is commonly used to measure the expected time for an event to occur (i.e. mine a block).Using this distribution makes calculating luck much simpler and it actually becomes more accurate as the Network Difficulty increases. At the current Network Difficulty it shouldn’t result in more than a millionth of a % in error.If that was hard to follow the next section should help you visualize it.Probability Density Function (PDF)Using the Erlang Distribution, the PDF indicates how probable it is that the Luck Statistic will be some arbitrary value. At any time the probability of the Luck Statistic being an exact number (i.e. 1.00000000000) is 0%. Rather the PDF can be used to specify the probability of the Luck Statistic falling within a particular range of values (i.e. below 1.0).For reference the formula can be found below.Formula for a PDF (Erlang Distribution)You can use R or python to calculate the PDF yourself. But a simpler way
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