Final answer:
The Poisson distribution is connected to the exponential distribution through their relationship with the timing of events and occurrence rates, with each being applicable under inverse circumstances.
Step-by-step explanation:
The relationship between the Poisson distribution and the exponential distribution is a fundamental concept in probability theory. If the time between successive events follows an exponential distribution with a mean of μ units of time and times between events are independent, then the number of events per unit time follows a Poisson distribution with a mean λ = 1/μ. Conversely, if the number of events per unit time follows a Poisson distribution, then the time between events follows an exponential distribution.
Examples of Poisson experiments include the number of misspelled words in a book or the number of occurrences within a fixed time interval. For the exponential distribution, examples might include the duration of phone calls or the lifetime of a product. Both distributions are useful for different types of random events and are interconnected.
To approximate a binomial distribution with a Poisson distribution, the probability of success should be low (p ≤ .05) and the number of trials should be high (n ≥ 20). This approximation is related to both distributions' tendency to deal with rare events over many trials or long time intervals.