Suppose p is the number of cars per minute passing through a certain road…
2013
Suppose p is the number of cars per minute passing through a certain road junction between 5 PM and 6 PM, and p has a Poisson distribution with mean 3. What is the probability of observing fewer than 3 cars during any given minute in this interval?
Answer: C. 17 / (2e3) — We want P(X < 3) for X ~ Poisson(mean = 3), so compute P(X=0)+P(X=1)+P(X=2). P(X=0) = e^-3 * 3^0 / 0! = e^-3 ≈ 0.0497871 P(X=1) = e^-3 * 3^1 / 1! = 3 e^-3 ≈…
- A.
8 / (2e3)
- B.
9 / (2e3)
- C.
17 / (2e3)
- D.
26 / (2e3)
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Correct answer: C
We want P(X < 3) for X ~ Poisson(mean = 3), so compute P(X=0)+P(X=1)+P(X=2).
P(X=0) = e^-3 * 3^0 / 0! = e^-3 ≈ 0.0497871
P(X=1) = e^-3 * 3^1 / 1! = 3 e^-3 ≈ 0.1493613
P(X=2) = e^-3 * 3^2 / 2! = (9/2) e^-3 = 4.5 e^-3 ≈ 0.2240420
Sum: P(X<3) = e^-3(1 + 3 + 9/2) = (17/2) e^-3 = 17/(2 e^3) ≈ 0.42319
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