<\/span><\/h2>\nThe <\/span>Poisson distribution formula<\/span> (more specifically, the probability mass function of the <\/span>Poisson distribution<\/span>) is as follows:\u00a0<\/span>\u00a0<\/span><\/p>\nP(x) = (e<\/span>\u2013 \u03bb<\/span> \u03bb<\/span>x<\/span>)\/x!<\/span>\u00a0<\/span><\/p>\nwhere<\/span>\u00a0<\/span><\/p>\ne is the base of the logarithm (Euler\u2019s constant, ~2.718)\u00a0<\/span>\u00a0<\/span><\/p>\nx is a Poisson random variable<\/span>\u00a0<\/span><\/p>\n\u03bb is an average rate of value<\/span>\u00a0<\/span><\/p>\n! is the factorial function<\/span>\u00a0<\/span><\/p>\nHere is a fictional example to explain the <\/span>Poisson distribution formula<\/span>:<\/span>\u00a0<\/span><\/p>\nIn a futuristic city, autonomous delivery drones are used to transport packages between various locations. The city is divided into sectors, each with its own network of drones managed by a central control system. The engineers responsible for maintaining the drone fleet are interested in analyzing the frequency of drone malfunctions per month in each sector. The average number of drone malfunctions per month in each sector follows a <\/span>Poisson distribution<\/span>. They decide to model this distribution to better understand the occurrence of malfunctions and plan preventive maintenance accordingly. For example, they find that in Sector 3, the average number of drone malfunctions per month is 5. Using the <\/span>Poisson distribution<\/span>, they can calculate the probability of one malfunction in a month in this sector as (2.718<\/span>\u2013 5<\/span> \u00d7 5<\/span>1<\/span>)\/1!\u00a0<\/span>\u00a0<\/span><\/p>\nwhere<\/span>\u00a0<\/span><\/p>\nx = 1 malfunction occurring in a month<\/span>\u00a0<\/span><\/p>\n\u03bb = average of 5 malfunctions a month<\/span>\u00a0<\/span><\/p>\ne = 2.718<\/span>\u00a0<\/span><\/p>\nUsing the <\/span>Poisson distribution formula<\/span>, the probability of a single drone malfunction occurring in Sector 3 is P(x = 1) = 0.03369. Using the <\/span>Poisson distribution<\/span>, the engineers can calculate the probability of different numbers of malfunctions occurring in a given month for each sector. This information allows them to anticipate potential issues, schedule maintenance activities proactively, and ensure that the drone network operates smoothly.<\/span>\u00a0<\/span><\/p>\n![\"\"](\"https:\/\/blog.researcher.life\/wp-content\/uploads\/2023\/03\/AAP-Banner-3-1024x410.png\")
<\/a><\/p>\n<\/span>Probability mass function graphs<\/span><\/b>\u00a0<\/span><\/span><\/h3>\nA probability density function is associated with continuous rather than discrete random variables. A probability mass function is different from this as it describes the probability of each possible value of a discrete random variable. That is, a probability mass function describes a discrete probability distribution. A <\/span>Poisson distribution<\/span> is a type of discrete probability distribution and can be visually depicted as a graph of the probability mass function.\u00a0<\/span><\/p>\n![\"\"](\"https:\/\/blog.researcher.life\/wp-content\/uploads\/2024\/04\/Poisson-Fig-1.png\")
\u00a0<\/span><\/p>\nFig. 1: Difference between<\/span><\/i> probability density function and probability mass function.[<\/span><\/i>1]<\/span><\/i>\u00a0<\/span><\/p>\nThe peak of this distribution is called the mode, the most probable number of events. When \u03bb is low, the distribution is right-skewed, that is, it is more stretched towards the right side of its peak than its left. As \u03bb increases, the distribution resembles normal distribution.<\/span>\u00a0<\/span>\u00a0<\/span><\/p>\n![\"\"](\"https:\/\/blog.researcher.life\/wp-content\/uploads\/2024\/04\/Poisson-Fig-2.png\")
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Fig. 2: Probability mass function for Poisson distribution.[<\/span><\/i>2]<\/span><\/i>\u00a0<\/span><\/p>\n![\"\"](\"https:\/\/blog.researcher.life\/wp-content\/uploads\/2024\/04\/Poisson-Fig-3.png\")
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Fig. 3: Graph of Poisson distribution or Poisson distribution curve.[<\/span><\/i>3]<\/span><\/i>\u00a0<\/span><\/p>\n<\/span>Poisson distribution table<\/span><\/b>\u00a0<\/span><\/span><\/h3>\nA <\/span>Poisson distribution table<\/span> tabulates the probabilities for a Poisson distribution. The probabilities can be calculated using the probability mass function of Poisson distribution. This table may be used to make it easier to calculate probabilities using the Poisson distribution. You can refer to the values from the table and substitute it in the <\/span>Poisson distribution formula<\/span><\/a> to obtain the value of the probability.<\/span>\u00a0<\/span><\/p>\n![\"\"](\"https:\/\/blog.researcher.life\/wp-content\/uploads\/2024\/04\/Poisson-Fig-4.png\")
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Table 1: Poisson probabilities for different values of <\/span><\/i>
![\"poisson](\"https:\/\/blog.researcher.life\/wp-content\/uploads\/2024\/04\/businesspeople-working-finance-accounting-analyze-financi-1.jpg\")
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