Question

2. (Poisson/Exponential/Gamma)

   Setup: The number of automobiles that arrive at a certain intersection per minute has a Poisson distribution with mean 4. Interest centers around the time that elapses before 8 automobiles appear at the intersection.

   Questions:

   (a) What is the probability that more than 3 minutes elapse before 8 cars arrive? (b) What is the probability that more than 1 minute elapses between arrivals?

Answer 1

The number of automobiles that arrive at a certain intersection per minute has a Poisson distribution with mean 4.
Interest centers around the time that elapses before 8 automobiles appear at the intersection
POSSION DISTRIBUTION
pmf of P.D is = f ( k ) = e-λ λx / x!
where   
λ = parameter of the distribution.
x = is the number of independent trials
I.
mean = λ
= 4

a.
the probability that more than 3 minutes elapse before 8 cars arrive
P( X < = 3) = P(X=3) + P(X=2) + P(X=1) + P(X=0)   
= e^-4 * 4 ^ 3 / 3! + e^-4 * 0 ^ 2 / 2! + e^-4 * 3 ^ 1 / 1! + e^-4 * 5 ^ 0 / 0!
= 0.43347,
P( X > 3) = 1 -P ( X <= 3) = 1 - 0.4335 = 0.56653

b.
the probability that more than 1 minute elapses between arrivals
P( X < = 1) = P(X=1) + P(X=0)   
= e^-4 * 4 ^ 1 / 1! + e^-4 * 0 ^ 0 / 0!
= 0.09158,
P( X > 1) = 1 -P ( X <= 1) = 1 - 0.0916 = 0.90842