If a Poisson random variable X has an average of 8.8, find:
0.9122
0.8695
0.9985
0.8124
0.0754
To find these probabilities, we can use the formula for the Poisson distribution:
P(X = k) = (e^(-λ) * λ^k) / k!
where λ is the average rate, in this case 8.8, and k is the number of occurrences.
1. P(X = 11): k = 11
P(X = 11) = (e^(-8.8) * 8.8^11) / 11! = (0.000164 * 3430675063.11) / 39916800 = 0.8759
2. P(X ≤ 10): to find this probability, we need to calculate the cumulative probability for X = 0, 1, 2, ..., 10
P(X ≤ 10) = Σ(P(X = k)) for k = 0 to 10 = 0.9122
3. P(X > 5): to find this probability, we can find 1 - P(X ≤ 5)
P(X > 5) = 1 - P(X ≤ 5) = 1 - Σ(P(X = k)) for k = 0 to 5 = 1 - (0.3755 + 0.2493 + 0.0915 + 0.0256 + 0.0057 + 0.0010) = 0.2674
4. P(X = 8): k = 8
P(X = 8) = (e^(-8.8) * 8.8^8) / 8! = (0.000173 * 2176782336) / 40320 = 0.0127
5. P(X = 2): k = 2
P(X = 2) = (e^(-8.8) * 8.8^2) / 2! = (0.000127 * 77.44) / 2 = 0.0049
So the answers are:
1. 0.8759
2. 0.9122
3. 0.2674
4. 0.0127
5. 0.0049