If a Poisson random variable X has an average of 5.5 find:
5.8
6.2
3.8
4.5
5.5
To find the probability of X being equal to a specific value, you can use the Poisson probability mass function formula:
P(X = k) = (e^-λ * λ^k) / k!
where:
- P(X = k) is the probability of X being equal to a specific value k
- e is the base of the natural logarithm (approximately equal to 2.71828)
- λ is the average rate at which events occur (in this case, 5.5)
- k is the specific value for which you want to calculate the probability
- k! is the factorial of k
Now, let's calculate the probabilities for each of the values:
1. For X = 5.8:
P(X = 5.8) = (e^-5.5 * 5.5^5.8) / 5.8!
P(X = 5.8) ≈ (0.0048 * 172.2784) / 407.3058
P(X = 5.8) ≈ 0.002034
2. For X = 6.2:
P(X = 6.2) = (e^-5.5 * 5.5^6.2) / 6.2!
P(X = 6.2) ≈ (0.0048 * 214.2853) / 480.3946
P(X = 6.2) ≈ 0.002139
3. For X = 3.8:
P(X = 3.8) = (e^-5.5 * 5.5^3.8) / 3.8!
P(X = 3.8) ≈ (0.0048 * 101.6353) / 29.4660
P(X = 3.8) ≈ 0.016581
4. For X = 4.5:
P(X = 4.5) = (e^-5.5 * 5.5^4.5) / 4.5!
P(X = 4.5) ≈ (0.0048 * 83.9204) / 24
P(X = 4.5) ≈ 0.016588
5. For X = 5.5:
P(X = 5.5) = (e^-5.5 * 5.5^5.5) / 5.5!
P(X = 5.5) ≈ (0.0048 * 163.4012) / 120
P(X = 5.5) ≈ 0.006534
Therefore, the probabilities for each value are approximately:
- P(X = 5.8) ≈ 0.002034
- P(X = 6.2) ≈ 0.002139
- P(X = 3.8) ≈ 0.016581
- P(X = 4.5) ≈ 0.016588
- P(X = 5.5) ≈ 0.006534