One model for plant competition assumes that there is a zone of resource depletion around each plant seedling. Depending on the size of the zones and the density of the plants, the zones of resource depletion may overlap with those of other seedlings in the vicinity. When the seeds are randomly dispersed over a wide area, the number of neighbors that a seedling may have usually follows a Poisson distribution with a mean equal to the density of seedlings per unit area. Suppose that the density of seedlings is four per square meter (m2).
a. What is the probability that a given seedling has no neighbors within 1 m2?
b. What is the probability that a seedling has at most three neighbors per m2?
c. What is the probability that a seedling has five or more neighbors per m2?
d. Use the fact that the mean and variance of a Poisson random variable are equal to find the proportion of neighbors that would fall into the interval µ ± 2σ. Comment on this result.