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I saw the following problem and thought Poisson would be the correct distribution, but the solution manual says otherwise.

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My textbook says an event is rare if the number of trials is at least 50, while $np \leq 5$. Here $np = 20*.05 = 1 \leq 5$.

I am guessing that the reason we can't use Poisson is because $n < 50$?

Is the above rule of thumb how I should proceed in general? ($n \geq 50$ and $np < 5$)

Joseph DiNatale
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  • Poisson distributions yields an approximation of the exact value of the desired probability, based on the binomial distribution. Here, the exact value is the sum of just three terms so, why not compute this sum? – Did Mar 08 '15 at 15:27
  • The crucial assumption that gives the binomial is that there cannot be more than $1$ hurricane in a year. – André Nicolas Mar 08 '15 at 15:43

1 Answers1

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To sum up and perhaps extend a bit:

Strictly speaking, the number N of hurricanes in 20 years is binomial with $n = 20$, $p = .05$, and $\mu = np = 1$. So $P\{N < 3\} = P\{N \le 2\} = 0.9245163,$ based on sufficiently simple computation (as shown) for a multiple-question, multiple-choice exam.

However, Poisson with $\lambda = \mu = np = 1$ gives 0.9196986 as a serviceable approximation, which would also lead to the correct 2-place answer. Also, possible answers provided are far enough apart that the Poisson approximation would not have had to be quite this good to guess the correct answer. But the Poisson computation involves an exponential and may not be as easy to compute as the binomial.

The Poisson would apply about as well as an approximation for the assumption that there can be at most one hurricane in a 2-year period with probability 1/10 so that N ~ BINOM(10, .1) [for which the exact probability of two or fewer events is 0.9298092]; alternatively, for an assumption that there can be at most one hurricane every six months with probability .025 so that N ~ BINOM(40, .025) [exact probability 0.9220516].

All that is required for a reasonable Poisson result is to have data supporting $\lambda = 1$ hurricane per 20 years on average and to be willing to believe that hurricanes will continue to occur as random events across time at that rate. (The restriction against multiple hurricanes within a year is necessary for an exact binomial computation, but seemingly not for a useful actuarial model.)

BruceET
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