# Binomial vs Poisson Distributions

• jmm
In summary, the Vancouver Island Marmot population and Grizzly bear populations in Banff National Park can be best modeled by binomial and Poisson distributions, respectively. Additionally, the number of butterflies landing in a square meter quadrat, the incidence of prostate cancer in men between the ages of 25-35, and the annual number of elephant attacks on humans in Serengeti National Park can also be modeled by Poisson distributions. The number of female frogs drawn from an artificial pond and the number of male spiders eaten after mating can be modeled by binomial distributions. However, the number of successful attacks by an Osprey on fish in a 0.5 km stretch of the Bow River may not be accurately modeled by either
jmm

## Homework Statement

I was given two problems and required to calculate some statistics/parameters for them. They are:

1) The Vancouver Island Marmot is one of Canada’s most endangered species; there are currently only 63 animals left on the Island. To maintain the population, geneticists believe it is important to keep the population above 60 animals to avoid inbreeding depression. This winter is anticipated to be particularly harsh, and it is expected that the over wintering probability of death could be 0.08.

2) You are a wildlife biologist surveying Grizzly bear populations in Banff National Park. Based on your previous research, you know that there is an average of 0.34 bears/km2 in the park.

The calculations aren't what's really troubling me, I'm just having a hard time deciding whether these would be best modeled by a Poisson or binomial distribution.

n/a

## The Attempt at a Solution

I think a Poisson distribution would fit best for both because they describe occurrences that take place per unit of time or space, but I have trouble with deciding.

Thanks.

Edit: The more I think about it, I'm starting to think that number 1) might just be a binomial distribution. There's 63 trials each with a probability of 'success' of 0.08. Almost like a weighted coin flip.

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jmm said:

## Homework Statement

I was given two problems and required to calculate some statistics/parameters for them. They are:

1) The Vancouver Island Marmot is one of Canada’s most endangered species; there are currently only 63 animals left on the Island. To maintain the population, geneticists believe it is important to keep the population above 60 animals to avoid inbreeding depression. This winter is anticipated to be particularly harsh, and it is expected that the over wintering probability of death could be 0.08.

2) You are a wildlife biologist surveying Grizzly bear populations in Banff National Park. Based on your previous research, you know that there is an average of 0.34 bears/km2 in the park.

The calculations aren't what's really troubling me, I'm just having a hard time deciding whether these would be best modeled by a Poisson or binomial distribution.

n/a

## The Attempt at a Solution

I think a Poisson distribution would fit best for both because they describe occurrences that take place per unit of time or space, but I have trouble with deciding.

Thanks.

Edit: The more I think about it, I'm starting to think that number 1) might just be a binomial distribution. There's 63 trials each with a probability of 'success' of 0.08. Almost like a weighted coin flip.

Your "edit" remark is correct, assuming independence of individual deaths (which one could argue for or against). The Poisson distribution would seem appropriate for 2), at least provided that there are not very strong interactions between different bears. (If bears like to avoid each other's territories, their occupancies of spaces are not independent and the results might not be Poisson. However, that is a matter that must be taken up with wildlife biologists/ecologists to get real facts.)

Thanks so much; that makes sense. This is for a quantitative biology class which is just a fancy name for stats 101 so we can definitely assume it's all independent. I've got some other questions while I'm at it: again I need to decide whether these would best modeled by binomial or Poisson distributions:

a. The number of butterflies landing in a square meter quadrat in an alpine field over the course of an hour.
b. The number of female frogs out of 10 drawn one by one from an artificial pond containing 25 female and 25 male frogs, if the frogs are replaced and mixed after each draw.
c. The incidence of prostate cancer in men between the ages of 25-35.
d. The number of male spiders eaten by the female after mating out of the 52 mating events observed.
e. Annual number of elephant attacks on humans in Serengeti National Park.
f. The number of successful attacks by an Osprey on fish in a 0.5 km stretch of the Bow River.

a. Poisson
b. Binomial
c. Poisson
d. Binomial
e. Poisson
F. Poisson - however I'm leaning towards binomial now after re-reading. I don't know if it's saying successful attacks/km or successful attacks per some number of attacks.

I'd appreciate you checking these when you have the time.

jmm said:
Thanks so much; that makes sense. This is for a quantitative biology class which is just a fancy name for stats 101 so we can definitely assume it's all independent. I've got some other questions while I'm at it: again I need to decide whether these would best modeled by binomial or Poisson distributions:

a. The number of butterflies landing in a square meter quadrat in an alpine field over the course of an hour.
b. The number of female frogs out of 10 drawn one by one from an artificial pond containing 25 female and 25 male frogs, if the frogs are replaced and mixed after each draw.
c. The incidence of prostate cancer in men between the ages of 25-35.
d. The number of male spiders eaten by the female after mating out of the 52 mating events observed.
e. Annual number of elephant attacks on humans in Serengeti National Park.
f. The number of successful attacks by an Osprey on fish in a 0.5 km stretch of the Bow River.

a. Poisson
b. Binomial
c. Poisson
d. Binomial
e. Poisson
F. Poisson - however I'm leaning towards binomial now after re-reading. I don't know if it's saying successful attacks/km or successful attacks per some number of attacks.

I'd appreciate you checking these when you have the time.

I agree with your answers, except maybe for f). If there are few fish in that stretch to begin with, and if the first few osprey are successful, later osprey are dealing with a reduced fish population and have a lower success probability than the first few osprey. That can make the overall results non-Poisson---and non-binomial as well. However, in the spirit of a first course in Prob and Stats, let's forget about real-life complications and just go with Poisson as a first approximation.

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Thanks again!

Remember that the Poisson distribution is just a limiting case of the binomial distribution, when the population of potential successes is very large and the probability of an individual success correspondingly small. In b and d you have specified smallish populations, so binomial is right. In a, c and e the population sizes are unknown, but presumably large, so Poisson.
What makes f tricky is that the population size (total number of attacks) is unknown, but is perhaps not very large. (Ray, I think the question relates to one specific Osprey, so you don't need to worry about predations by others.) Worse, I suspect the number of successful attacks will be more predictable than the total number of attacks. An Osprey can only eat so much. So, paradoxically, I doubt a Poisson distribution is appropriate for catching these poissons. Gaussian anyone?

## What is the difference between a binomial and Poisson distribution?

A binomial distribution is used to model the probability of a certain number of successes in a fixed number of independent trials, while a Poisson distribution is used to model the probability of a certain number of events occurring in a fixed time period or space.

## What are the key characteristics of a binomial distribution?

A binomial distribution has a fixed number of trials, a constant probability of success for each trial, and each trial is independent of the others. Additionally, the outcomes of each trial are dichotomous (success or failure).

## How is the mean calculated for a binomial distribution?

The mean for a binomial distribution is equal to the product of the number of trials (n) and the probability of success (p). This can be written as μ = np.

## What is the role of lambda in a Poisson distribution?

Lambda (λ) is the average rate of events occurring in a fixed time period or space. It is used to calculate the mean and variance of a Poisson distribution, which is μ = λ and σ² = λ, respectively.

## When should a binomial distribution be used over a Poisson distribution?

A binomial distribution should be used when there are a fixed number of trials and a constant probability of success for each trial. A Poisson distribution should be used when the number of events occurring in a fixed time period or space is of interest and the events are independent of each other.