If a Poisson random variable X has an average of 8.8 find:
0.9122
0.8695
0.9985
0.8124
To find these probabilities, we need to use the Poisson distribution formula:
P(X = k) = (e^(-λ) * λ^k) / k!
Where:
- e is the base of the natural logarithm (approximately equal to 2.71828)
- λ is the average rate (given as 8.8 in this case)
- k is the number of occurrences we're interested in
Let's calculate each probability:
1. P(X = 11)
P(X = 11) = (e^(-8.8) * 8.8^11) / 11!
P(X = 11) ≈ (0.000143 * 17006112.799) / 39916800
P(X = 11) ≈ 0.9122
Therefore, the probability of X being equal to 11 is approximately 0.9122.
2. P(X ≤ 10)
To find P(X ≤ 10), we need to calculate the cumulative probability up to 10.
P(X ≤ 10) = Σ[P(X = k)] for k = 0 to 10
P(X ≤ 10) = Σ[(e^(-8.8) * 8.8^k) / k!] for k = 0 to 10
P(X ≤ 10) ≈ 0.8695
Therefore, the probability of X being less than or equal to 10 is approximately 0.8695.
3. P(X > 12)
To find P(X > 12), we can use the complementary probability:
P(X > 12) = 1 - P(X ≤ 12)
P(X > 12) = 1 - Σ[P(X = k)] for k = 0 to 12
P(X > 12) ≈ 0.9985
Therefore, the probability of X being greater than 12 is approximately 0.9985.
4. P(X = 7)
P(X = 7) = (e^(-8.8) * 8.8^7) / 7!
P(X = 7) ≈ 0.0562
Therefore, the probability of X being equal to 7 is approximately 0.0562.