{"id":1528,"date":"2024-05-13T16:16:22","date_gmt":"2024-05-13T20:16:22","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/?post_type=chapter&p=1528"},"modified":"2024-06-12T14:46:21","modified_gmt":"2024-06-12T18:46:21","slug":"an-introduction-to-poisson-distributions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/1130sandbox\/chapter\/an-introduction-to-poisson-distributions\/","title":{"raw":"Distribution Properties and Probabilities of Exactly 'X' Events","rendered":"Distribution Properties and Probabilities of Exactly ‘X’ Events"},"content":{"raw":"
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Learning Objectives<\/p>\r\n\r\n<\/header>\r\n

\r\n\r\nUnderstand what it means for events to follow a Poisson distribution and calculate the probability of [latex]x[\/latex] events occurring per time period using the Poisson Probability Mass Function<\/a>.\r\n\r\n<\/div>\r\n<\/div>\r\n

Five Properties of Poisson Distributions<\/h2>\r\n
    \r\n \t
  1. There are only two possible outcomes for events \u2013 they can occur or not occur.<\/li>\r\n \t
  2. Each event must be random and independent of the others.<\/li>\r\n \t
  3. The occurrences must be uniformly distributed (see more in the uniform distributions<\/a> section).<\/li>\r\n \t
  4. Probabilities are calculated on the number of occurrences, [latex]x[\/latex], over a specific time interval.<\/li>\r\n \t
  5. An average number of occurrences over a given time period is denoted as [latex]\\lambda[\/latex].<\/li>\r\n<\/ol>\r\n

    Two Ways of Calculating the Probability of 'x<\/em>' Successes<\/h2>\r\nIf we want to calculate the probability of exactly [latex]x[\/latex] events occurring per time period, we can:\r\n
      \r\n \t
    1. Use the formula<\/span>: [latex]P(x) = \\frac{{\\lambda}^x e^{-\\lambda}}{x!}[\/latex]<\/li>\r\n \t
    2. Use Excel<\/span>: = POISSON.DIST<\/a><\/span>([latex]x[\/latex], [latex]\\lambda[\/latex], FALSE)<\/li>\r\n<\/ol>\r\n

      three examples of poisson distributions<\/h1>\r\nThree examples of events that follow a Poisson Distribution:\r\n
        \r\n \t
      1. The number of bad checks received by a bank per day.<\/li>\r\n \t
      2. The number of motor vehicle accidents in a busy street corner per week.<\/li>\r\n \t
      3. The number of shoppers arriving at a supermarket check-out counter per minute.<\/li>\r\n<\/ol>\r\n

        Calculating the probability of x events using the Formula (ViDEO)<\/h1>\r\nLet us better understand the Poisson formula by working through an example.\r\n

        Example 30.1.1<\/h2>\r\nProblem Setup<\/strong><\/span>: On a busy Friday evening, the average number of shoppers arriving at a supermarket check-out counter waiting to be served by a cashier is 6 per minute. The arrivals follow a Poisson distribution.\r\n\r\nQuestion<\/strong><\/span>: In any given minute, what is the probability that exactly 5 shoppers will arrive?\r\n\r\nWritten solution:<\/strong><\/span>\u00a0Let us look at the formula a little more carefully now.\r\n