Question

A local scientist surveyed numerous 10m x 10m plots in southern Queensland to investigate the presence of the troublesome invasive species, the cane toad. Alarmingly, she found a mean of 30 cane toads per plot. Under the assumption that the cane toads are randomly distributed and follow a poisson distribution answer the following

(a) What is the expected number of cane toads found per 1m^2 plot?

(b) What is the probability of finding no more than 3 toads in a 1m^2 plot?

(c) What is the probability of finding at least 2 toads in a 1m^2 plot? (3 marks

Answer 1

Solution:

Given: A local scientist surveyed numerous 10m x 10m plots in southern Queensland to investigate the presence of the troublesome invasive species, the cane toad. Thus area of plot =10m x 10m =100 m²  

She found a mean of 30 cane toads per plot.

X = Number of cane toad follows poisson distribution .

Part a) What is the expected number of cane toads found per 1m^2 plot?

We have mean of 30 cane toads per plot of area 100 m² .

Thus

Mean per  1 m² = 30 / 100 = 0.3 per  1 m²

the expected number of cane toads found per 1m^2 plot = 0.3

thus X ~ Poisson( ) per  1 m²

Part b) What is the probability of finding no more than 3 toads in a 1m^2 plot?

P( No more than 3 toads in a 1m^2 plot) = ..............?

Probability mass function of Poisson distribution is:

thus

Part c) What is the probability of finding at least 2 toads in a 1m^2 plot?

_{( Round final answer to specified number of decimal places)}