Grasshoppers are distributed at random in a large field according to a Poisson process with parameter a 5 2 per square yard. How large should the radius R of a circular sampling region be taken so that the probability of finding at least one in the region equals 0.99?

In this question, the Poisson distribution is used.

Poisson distribution:

In a Poisson distribution, the probability that X represents the number of successes of a random variable is given by the following formula:

In which

x is the number of sucesses

e = 2.71828 is the Euler number

is the mean in the given interval.

Parameter of 5.2 per square yard:

This means that , in which r is the radius.

How large should the radius R of a circular sampling region be taken so that the probability of finding at least one in the region equals 0.99?

We want:

Thus:

We have that:

Then

Thus, the radius should be of at least 0.89.

Another example of a Poisson distribution is found at