Understanding Poisson Distribution
The Poisson distribution is characterized by its mean (λ), which also equals the variance. The probability of observing exactly k events in a fixed interval is given by the formula:
P(X=k) = (λ^k * e^(-λ)) / k!
Where:
- λ = mean number of events
- e = Euler's number (approximately 2.71828)
- k = number of events
Given Conditions
We know:
- P(X=7) = 8 * P(X=9)
Using the formula for the Poisson distribution, we can express these probabilities:
- P(X=7) = (λ^7 * e^(-λ)) / 7!
- P(X=9) = (λ^9 * e^(-λ)) / 9!
Setting Up the Equation
From the given condition, we equate:
(λ^7 * e^(-λ)) / 7! = 8 * [(λ^9 * e^(-λ)) / 9!]
Simplifying the Equation
Cancelling e^(-λ) from both sides, we get:
(λ^7 / 7!) = 8 * (λ^9 / 9!)
This simplifies to:
9 * λ^7 = 8 * λ^9
Solving for λ
Rearranging gives:
9 = 8 * λ^2
Thus, λ^2 = 9/8, leading to λ = √(9/8) = 3/√8 = 3/(2√2).
The mean of the distribution, λ, is approximately 1.06.
Conclusion
Thus, the mean of the Poisson distribution in this scenario is approximately 1.06. Understanding these steps provides clarity on how to work with Poisson probabilities effectively.