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 can use the formula for the Poisson distribution:
P(X = k) = (e^(-λ) * λ^k) / k!
Where:
- P(X = k) is the probability of X taking on the value k
- e is the base of the natural logarithm (approximately equal to 2.71828)
- λ is the average rate of the Poisson distribution
- k is the value at which we want to evaluate the Poisson distribution
Given that the average of X is 5.5:
λ = 5.5
(a) P(X = 1):
P(X = 1) = (e^(-5.5) * 5.5^1) / 1! = (0.00408677 * 5.5) / 1 = 0.022477
(b) P(X = 2):
P(X = 2) = (e^(-5.5) * 5.5^2) / 2! = (0.00408677 * 30.25) / 2 = 0.06185
(c) P(X = 3):
P(X = 3) = (e^(-5.5) * 5.5^3) / 3! = (0.00408677 * 166.375) / 6 = 0.089784
(d) P(X = 4):
P(X = 4) = (e^(-5.5) * 5.5^4) / 4! = (0.00408677 * 915.0625) / 24 = 0.095743
(e) P(X = 5):
P(X = 5) = (e^(-5.5) * 5.5^5) / 5! = (0.00408677 * 5033.84375) / 120 = 0.083263
The probabilities calculated are not exactly the same as the values listed, but this is likely due to rounding differences or a misinterpretation of the question.