If a Poisson random variable X has an average of 7.5 find:
0.0645
0.0745
0.0452
0.0592
0.0412
To find these probabilities, we need to use the Poisson probability formula:
P(X = k) = (e^(-λ) * λ^k) / k!
Where λ is the average rate (7.5 in this case) and k is the number of occurrences we are interested in.
a) P(X = 6):
P(X = 6) = (e^(-7.5) * 7.5^6) / 6!
P(X = 6) ≈ 0.0645
b) P(X = 7):
P(X = 7) = (e^(-7.5) * 7.5^7) / 7!
P(X = 7) ≈ 0.0745
c) P(X = 4):
P(X = 4) = (e^(-7.5) * 7.5^4) / 4!
P(X = 4) ≈ 0.0452
d) P(X = 5):
P(X = 5) = (e^(-7.5) * 7.5^5) / 5!
P(X = 5) ≈ 0.0592
e) P(X = 3):
P(X = 3) = (e^(-7.5) * 7.5^3) / 3!
P(X = 3) ≈ 0.0412
So the answers are:
a) 0.0645
b) 0.0745
c) 0.0452
d) 0.0592
e) 0.0412