Problem 12E

Two-dimensional Poisson process. The number of plants of a certain species in a certain forest has a Poisson distribution with mean 10 plants per acre. The number of plants in T acres therefore has a Poisson distribution with mean 10T

a. What is the probability that there will be exactly 18 plants in a two-acre region?

b. What is the probability that there will be exactly 12 plants in a circle with radius 100 ft? (1 acre = 43,560 ft2.)

c. The number of plants of a different type follows a Poisson distribution with mean λ plants per acre, where λ is unknown. A total of 5 plants are counted in a 0.1 acre area. Estimate λ, and find the uncertainty in the estimate.

Solution

Step 1 of 3

Here given the no. of plants in forest has a poisson distribution with mean 10 per acre

Let X represents the no. of plants

The pmf of poisson distribution is P(x) =, x=0,1,2,............

Then Xpoisson(10)

a) Here we have to find the probability that there will be exactly 18 plants in 2 acre

For one acre 10 plants

For 2 acre 20 plants

Hence =20

Now P(x=18)=

=0.0844

Hence the probability that there will be exactly 18 plants in 2 acre is 0.0844