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I had this problem on my last midterm and received no credit for these parts.

We want E[# of trains]

lambda

Given 9:00-11:00 => 10 trains

<=>\int_{9}^{11}4c dr = 10 => c=5/4

\int_{13}^{15}4c dr = ? = 10 <=> E[# of trains] = 10

Sorry, latex typesetting was not working.

E[T]=1/lambda

mu=24

lambda

E[# of passengers]=mu(P(T \leq 12))=24(1-exp(-12lambda

P(#trains=12 & #buses=12 | time = 1 hr)

average 4 trains per hour

average 8 buses per hour

Now, I am stuck.

**1. Express trains arrive at Hiawatha station according to a Poisson process at rate 4 per hour, and independent of this, Downtown local buses arrive according to a Poisson process at rate 8 per hour.****a. Given that 10 Express trains arrive during the morning hours of 9:00-11:00 am, what is the***expected*number of Express trains that will arrive during the afternoon hours of 1:00-3:00pm**The attempt at a solution**We want E[# of trains]

lambda

_{trains}=4Given 9:00-11:00 => 10 trains

<=>\int_{9}^{11}4c dr = 10 => c=5/4

\int_{13}^{15}4c dr = ? = 10 <=> E[# of trains] = 10

Sorry, latex typesetting was not working.

**b. Suppose your friend arrives at the station and decides to take the first transportation that arrives. Given she takes an Express train (it arrives first), what is the expected amount of time she waited for it to arrive?****The attempt at a solution**E[T]=1/lambda

_{trains}=0.25=15minutes**c. Suppose each Downtown bus carries passengers, the number of which has a probability distribution with mean 24. Find the expected value of the number of passengers that ride the Downtown bus line during 12 hours.****The attempt at a solution**mu=24

lambda

_{bus}=8E[# of passengers]=mu(P(T \leq 12))=24(1-exp(-12lambda

_{bus})=24**d. What is the probability that a total of exactly 12 trains and buses arrive in a given 1-hour period?****The attempt at a solution**P(#trains=12 & #buses=12 | time = 1 hr)

average 4 trains per hour

average 8 buses per hour

Now, I am stuck.

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