Respuesta :
Answer:
a) 372640 / e^80
b) E(X) = 6,800,000 trees
c) [tex]pmf = \frac{40212^k * e^-^4^0^2^1^2}{k!}[/tex]
Step-by-step explanation:
Given:-
- The parameter (λ) = 80
- The random variable X:
          X ~ Po ( 80 )
Solution:-
a) P ( X ≤ 16 ):-
- We will use the pmf function for the poisson distribution to evaluate the asled probability as follows:
          [tex]P(X\leq 16) = \sum _{n=0}^{16}\:\frac{\left(λ\right)^n\cdot \left(e^{-λ}\right)}{n!}\\\\P(X\leq 16) = \sum _{n=0}^{16}\:\frac{\left(80\right)^n\cdot \left(e^{-80}\right)}{n!}\\\\P(X\leq 16) = \frac{372640}{e^{80}}[/tex]
b) If the forest covers 85,000 acres, what is the expected number of trees in the forest?
- From given data it is known that:
     per acre : E(X) = 80
     85,000 acres : E(X) = 85,000*(80)
                      = 6,800,000 trees per 85,000 acres
c) Suppose you select a point in the forest and construct a circle of radius 0.1 mile. Let X the number of trees within that circular region. What is the pmf of X?
Solution:-
- The value of the parameter ( λ ) is given for "acres". We will first convert acres to square miles.
          1 acre = 0.0015625 miles^2   Â
- So,
          λ = 80 trees / 0.0015625 miles^2 Â
          λ = 51,200 trees / miles^2      Â
- The area covered by the circular region is denoted by its radius r = 0.5 miles.
          A_circle = π*r^2
                 = π*(0.5)^2
                 = 0.78539 miles^2
- Using direct proportions we have:
         1 square mile  --------- >  51,200 trees
         0.78539 square mile ---> x trees
       =====================================
         x = 51,200*(0.78539) = 40,212 trees Â
       =====================================
- The random variable (X) follows the Poisson distribution with parameter (  λ = 40,212 trees / miles ) with pmf:
        [tex]pmf = \frac{40212^k * e^-^4^0^2^1^2}{k!}[/tex]
