If a Poisson random variable X has an average of 2.6 find:
0.3112
0.2876
0.2133
0.1889
0.1226
To find these probabilities, we can use the Poisson probability formula:
P(X = k) = e^(-λ) * λ^k / k!
Where:
P(X = k) is the probability of X taking a specific value k
e is Euler's number (approximately 2.71828)
λ is the average rate of events in a fixed interval (in this case, 2.6)
k is the number of events
1. P(X = 0) = e^(-2.6) * 2.6^0 / 0! = e^(-2.6) ≈ 0.0739
2. P(X = 1) = e^(-2.6) * 2.6^1 / 1! = 2.6 * e^(-2.6) ≈ 0.1908
3. P(X = 2) = e^(-2.6) * 2.6^2 / 2! = 6.76 * e^(-2.6) ≈ 0.2495
4. P(X = 3) = e^(-2.6) * 2.6^3 / 3! = 17.576 * e^(-2.6) ≈ 0.2133
5. The probabilities in descending order are 0.2495, 0.2133, 0.1908, 0.0739
Therefore, the probabilities are:
0.3112 = P(X ≤ 3) = 0.2495 + 0.2133 + 0.1908 + 0.0739
0.2876 = P(X ≤ 2) = 0.2495 + 0.2133 + 0.1908
0.2133 = P(X = 3)
0.1889 = P(X ≤ 1) = 0.0739 + 0.1908
0.1226 = P(X = 0) = 0.0739