EXAMPLE 24Poisson approximation to the binomial distribution

Two percent of online e-commerce transactions are fraudulent.5 (a) In a sample of 100 online e-commerce transactions, approximate the probability that three fraudulent transactions occur. (b) Measure the accuracy of the approximation.

Solution

  1. We first verify that the requirements are met. The number of trials is . The probability of “success” (that is, fraud) on any particular transaction is , so that . The requirements are met. Next, we find the mean of the Poisson distribution used to approximate the binomial distribution: . Then the probability that fraudulent transactions occur is

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    The probability is 0.1804 that three fraudulent transactions occur in the sample of 100 online e-commerce transactions.

  2. How good an approximation is this? Using the more accurate binomial distribution, we get , meaning that the Poisson approximation is accurate in this instance to within of the binomial probability.

NOW YOU CAN DO

Exercises 19–22.