# When is calculating probability based on a poisson distribution preferred over a binomial distribution?

I'm trying to understand why a poisson distribution may be preferred over a binomial one when modeling binary cases.

1. Is there a case where you can't use a binomial distribution to solve a problem that can be solved by a poisson distribution? For example: The average rate for fishing is 2 fish caught per day. What is the probability that 1 fish a day is caught? This can be modeled by a binomial distribution if you define a trial as each hour and the probability of success is approximately 2 fish/24 hour = 0.0833 per hour. Assuming there are 24 trials, the answer calculated from Binomial(24, 0.0833) is approximately the same as the poisson.

2. When is a binomial distribution unreasonable to calculate computationally? I understand that calculating a probability based on a poisson distribution is less computationally expensive than calculating it based on a binomial distribution. What are the practical applications of Poisson where computational time actually makes large impact?

• Read up on rare events. Jan 15, 2015 at 17:34
• How you phrase the first question is amusing because it flips the title on its head. You are asking why not use the Binomial to approximate a Poisson distribution! Here we have a situation where you seem to be advocating approximating the correct distribution with something computationally more difficult to evaluate. Why is that?
– whuber
Jan 15, 2015 at 17:38
• (When modeling fishing, the Poisson distribution sounds like it would be particularly applicable. Francophones will understand what I mean :-).)
– whuber
Jan 15, 2015 at 19:40

Thus, when performing regression on continous (time-series) data such as fish caught or neuronal spiking, the advantage of using Poisson regression (such as w/ a GLM w/ an exponential link function) over bernoulli regression (such as logistic regression) is that once you have the estimated poisson rate function $r(t)$, you can always recreate the bernoulli firing probability in every bin as $P(event-in-\Delta_t)=\int_{\Delta_t} r(t)dt$