Final answer:
The Poisson probability distribution is used to find the probability of a number of events occurring given an average rate of occurrence with λ being the mean. The formula P(X=k) = (e^-λ * λ^k) / k! calculates the exact probability for various λ values.
Step-by-step explanation:
To solve these problems, we use the Poisson probability distribution formula:
where λ (lambda) is the average rate of occurrence, k is the number of occurrences, and e is approximately 2.71828 (Euler's number).
a. With λ=2.5, find P(X=6):
P(X=6) = (e-2.5 * 2.56) / 6! ≈ 0.0668
b. With λ=8.0, find P(X=1):
P(X=1) = (e-8.0 * 8.01) / 1! ≈ 0.0003
c. With λ=0.5, find P(X=2):
P(X=2) = (e-0.5 * 0.52) / 2! ≈ 0.0758
d. With λ=3.7, find P(X=9):
P(X=9) = (e-3.7 * 3.79) / 9! ≈ 0.0320
Note that to find these probabilities, we can use a scientific calculator or statistical software to compute these values more accurately.