If a Poisson random variable X has an average of 5.5, find:
0.2136
0.1571
0.0894
0.0956
0.2654
To find these probabilities, we will need to use the Poisson probability formula, which is:
P(X = k) = (e^(-λ) * λ^k) / k!
Where λ represents the average rate of occurrence (in this case 5.5) and k is the value we are interested in finding the probability for.
1. P(X = 2)
P(X = 2) = (e^(-5.5) * 5.5^2) / 2!
P(X = 2) = (0.004086771 * 30.25) / 2
P(X = 2) = 0.061848457 / 2
P(X = 2) = 0.0309242285
2. P(X = 3)
P(X = 3) = (e^(-5.5) * 5.5^3) / 3!
P(X = 3) = (0.004086771 * 166.375) / 6
P(X = 3) = 0.6807677343 / 6
P(X = 3) = 0.1134612891
3. P(X = 6)
P(X = 6) = (e^(-5.5) * 5.5^6) / 6!
P(X = 6) = (0.004086771 * 10128.625) / 720
P(X = 6) = 41.38513214 / 720
P(X = 6) = 0.0573965175
4. P(X = 7)
P(X = 7) = (e^(-5.5) * 5.5^7) / 7!
P(X = 7) = (0.004086771 * 70204.125) / 5040
P(X = 7) = 286.83057475 / 5040
P(X = 7) = 0.0568826787
5. P(X = 10)
P(X = 10) = (e^(-5.5) * 5.5^10) / 10!
P(X = 10) = (0.004086771 * 305246.4424) / 3628800
P(X = 10) = 1246.804866 / 3628800
P(X = 10) = 0.0003431325
The probabilities given in the question do not match with the values calculated above.