The potholes on a major arterial in Seattle occur at the rat

The potholes on a major arterial in Seattle occur at the rate of 3.4 per mile. Compute the following: What is the probability that there are exactly five potholes over 1 mile of randomly selected arterial road? Interpret this result. What is the probability that there are exactly five potholes over 2 miles of randomly selected arterial road? Interpret this result. What is the probability that there are fewer than five potholes over 2 miles of randomly selected arterial road? Interpret this result.: It is thought that there are 1100 moose in the Yellowstone Park moose population. Last year 50 moose were captured and tagged. Six months later 200 moose are captured. Define the RV?X, to be the number of tagged moose in the group of 200 most recently captured moose. We will assume that all moose are still living and that the population total has not changed. Name the probability distribution that can be appropriately used to find probabilities of X. What is the probability that 25 of the most recently captured moose are tagged?

Solution

Possion Distribution
PMF of P.D is = f ( k ) = e- x / x!
Where   
= parameter of the distribution.
x = is the number of independent trials
Seattle occur at the rate of 3.4
a)
P( X = 5 ) = e ^-3.4 * 3.4^5 / 5! = 0.1264

b)
Seattle occur at the rate of fr 2 mile is 3.4*2 = 6.8
P( X = 5 ) = e ^-6.8 * 6.8^5 / 5! = 0.1349

c)
P( X < 5) = P(X=4) + P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= e^-6.8 * 5 ^ 4 / 4! + e^-6.8 * ^ 3 / 3! + e^-6.8 * ^ 2 / 2! + e^-6.8 * ^ 1 / 1! + e^-6.8 * ^ 0 / 0!
= 0.192