Question
If a Poisson random variable X has an average of 2.6, find:
0.3112
0.2876
0.2133
0.1889
0.1226
0.3112
0.2876
0.2133
0.1889
0.1226
Answers
GPT 3.5
The probabilities P(X=k) for a Poisson random variable are given by the formula:
P(X=k) = e^(-λ) * λ^k / k!
Where λ is the average value. In this case, λ = 2.6.
a) P(X=0) = e^(-2.6) * 2.6^0 / 0! = e^(-2.6) ≈ 0.0732
b) P(X=1) = e^(-2.6) * 2.6^1 / 1! = 2.6 * e^(-2.6) ≈ 0.1895
c) P(X=2) = e^(-2.6) * 2.6^2 / 2! = 3.38 * e^(-2.6) ≈ 0.4904
d) P(X=3) = e^(-2.6) * 2.6^3 / 3! = 10.39 * e^(-2.6) ≈ 0.8003
e) P(X=4) = e^(-2.6) * 2.6^4 / 4! = 27.06 * e^(-2.6) ≈ 0.9455
From the above calculations, the probability closest to 0.3112 is option a) P(X=0) ≈ 0.0732.
P(X=k) = e^(-λ) * λ^k / k!
Where λ is the average value. In this case, λ = 2.6.
a) P(X=0) = e^(-2.6) * 2.6^0 / 0! = e^(-2.6) ≈ 0.0732
b) P(X=1) = e^(-2.6) * 2.6^1 / 1! = 2.6 * e^(-2.6) ≈ 0.1895
c) P(X=2) = e^(-2.6) * 2.6^2 / 2! = 3.38 * e^(-2.6) ≈ 0.4904
d) P(X=3) = e^(-2.6) * 2.6^3 / 3! = 10.39 * e^(-2.6) ≈ 0.8003
e) P(X=4) = e^(-2.6) * 2.6^4 / 4! = 27.06 * e^(-2.6) ≈ 0.9455
From the above calculations, the probability closest to 0.3112 is option a) P(X=0) ≈ 0.0732.