Greetings I have a Bernoulli process and a Poisson process question as follows:
You are waiting at a bus stop and there are two kinds of buses stopping there: Bus A and Bus B. Each arriving bus is Bus A with probability 0.75, independenly from one arrival to the next. Note that we can model this as a Bernoulli process.
a. You would like to take Bus B, and show up at the bus stop at a random time. How many buses on average will arrive before you can get on one?
b. Find the expected number of Bus A arrivals between two consecutive Bus B arrivals?
c. You arrive at a random time. What is the expected number of Bus A arrivals between Bus B you get on and the previous Bus B?
The other question is as follows:
John receives text messages on his phone according to a Poisson process with a rate of 1 message per 20 minutes.
a. What is the probability that he gets at least one message in a duration of 5 minutes?
b. John woke up at 10:00 in the morning. What is the expected time at which he has received his first text message?
c. Find the moment generating function of X, the total duration between the first and the third messages.
Since I missed some lectures I am particularly weak at these topics so any help is appreciated. Thanks in advance.