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 
.)
c. The number of plants of a different type follows a Poisson distribution with mean ? plants per acre, where ? is unknown. A total of 5 plants are counted in a 0.1 acre area. Estimate ?. and find the uncertainty in the estimate.
(a)Let X be the number of plants in a two-acre region. Then X ~ Poisson(20).
(b) The area of the circle is 10,000 = 31,415.9 
. In units of acres, the area is 31,415.9/43,560 = 0.72121 acres. Let X be the number of plants in the circle. Then X ~ Poisson(7.2121).
(c) Let Y represent the number of plants observed in 0.1 acres. Let ? represent the true concentration in plants per acre.Then Y ~ Poisson(0.1? ). The observed value of Y is 5. The estimated concentration is ? = 5/0.1=50 .
The uncertainty is
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