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Understanding Poisson Distribution
The Poisson distribution is defined by its parameter λ (lambda), which represents the average rate (mean) of occurrence of an event. The probabilities for the number of occurrences (k) are given by:
\[ P(k; \lambda) = \frac{e^{-\lambda} \lambda^k}{k!} \]
Where:
- \( P(k; \lambda) \) is the probability of observing \( k \) events.
- \( e \) is the base of the natural logarithm.
- \( k! \) is the factorial of \( k \).
Given Frequency Terms
We are provided with the second and third frequency terms of the Poisson distribution:
- Frequency of the 2nd term (k=1): 100
- Frequency of the 3rd term (k=2): 80
Calculating λ
To find λ, we can set up the equations based on the frequencies:
1. For k=1:
\[ P(1; \lambda) = \frac{e^{-\lambda} \lambda^1}{1!} = 100 \]
2. For k=2:
\[ P(2; \lambda) = \frac{e^{-\lambda} \lambda^2}{2!} = 80 \]
From these equations, we can express \( P(1) \) in terms of \( P(2) \):
\[
P(2) = \frac{\lambda}{2} P(1) \rightarrow 80 = \frac{\lambda}{2} \cdot 100
\]
Solving for λ:
\[
\lambda = \frac{80 \cdot 2}{100} = 1.6
\]
Calculating the Next Frequency Term
Now, we need to find the frequency of the 4th term (k=3):
Using the Poisson formula for k=3:
\[
P(3; 1.6) = \frac{e^{-1.6} (1.6)^3}{3!}
\]
Calculating the value:
- \( e^{-1.6} \approx 0.2019 \)
- \( (1.6)^3 = 4.096 \)
- \( 3! = 6 \)
Thus,
\[
P(3; 1.6) \approx 0.2019 \cdot \frac{4.096}{6} \approx 0.1371
\]
To find the frequency term:
\[
\text{Frequency for k=3} \approx 0.1371 \times \text{Total Frequency}
\]
Since the total frequency can be deduced from previous terms, the expected frequency for k=3 is approximately:
\[
\text{Frequency} \approx 0.1371 \times (100 + 80 + \text{other terms})
\]
Conclusion
By calculating the above, we can estimate the next frequency term of the Poisson distribution based on the given data.