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Poisson Experiment
Poisson Experiment in Fisheries and Seafood
Counting the Number of Fish Caught in a Fixed Period
Description of The Poisson Experiment:
Number of events occurring within a fixed interval of time or space: The number of
fish caught by a fishing boat in a day.
There is a known average rate at which events happen. For example, based on past
information, the average number of fish caught by a boat is seen to be 20 every day.
It is consequential of the time elapsed since the last event: every capture is
independent of the previous one. Catching a fish now does not affect the probability
of catching another fish in the next instant.
How the Selected Experiment Meets the Poisson Characteristics:
1. Fixed Interval of Time or Space:
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The experiment entails enumerating how many fish were caught within a time frame
of a day.
2. Fair Average Rate:
Overall, the average rate of fish per day, confirmed by continuous historical
observation over the period, is 20.
3. Independence of Events:
In basins, catching a fish is a completely random process, and consecutive catches are
independent of each other (Julia et al., 2021). In this case, the probability of catching a
fish at any time during the day is fixed.
Poisson Distribution as an Approximation to the Binomial Distribution
Explanation:
The Poisson distribution may be used to approximate the binomial probability distribution
when the number of trials, n, is large and the probability of access, p, is small, such that np is
a moderate-size number. Poisson distribution is an important tool in making approximation in
different studies (Tang & Tang, 2023). This follows because the Poisson distribution is, in
fact, derived from the Binomial distribution for large arises having small individual
probability, and the result is easy to work with, ideal when exact calculations of the Binomial
pro Eligibility distribution would be especially tedious.
Example:
Suppose we are monitoring the count of bycatch events (species caught besides the most
targeted) at a large fishery. Let's say we have:
n = 1000 fishing trips in a year.
p = 0.01 probability of a bycatch incident per trip.
Using the binomial distribution, we would calculate the probability of k bycatch incidents is:
P
(
X=k
)
=
(
1000
k
)
pk
(
1−p
)
1000−k
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Given n is large and p is small, the product
λ=np=1000 ×0.01=10
The Poisson distribution can approximate this binomial distribution:
P
(
X=k
)
=λke−λ
k !
For k=5k, the probability of 5 bycatch incidents is:
P
(
X=5
)
=105e−10
5!=0.0378
Using the Poisson distribution simplifies the calculations compared to the binomial,
especially for large n, providing a practical and efficient approximation.
References
Julia, M., Muñoz-Mas, R., Machado, J., Carolina Mendes Muniz, Santos, Emili García-
Berthou, & Luiz Carlos Gomes. (2021). Effects of reservoir cascades on diversity,
distribution, and abundance of fish assemblages in three Neotropical basins. Science
of the Total Environment, 778, 146246–146246.
https://doi.org/10.1016/j.scitotenv.2021.146246
Tang, W., & Tang, F. (2023). The Poisson Binomial Distribution— Old & New. Statistical
Science, 38(1). https://doi.org/10.1214/22-sts852
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