00:01
For this problem, we are told that students arrive according to a poisson process with a rate of 12 per hour.
00:08
So that means in this poisson process, our arrival rate, or our parameter lambda, is equal to 12.
00:17
Now remember, this is per hour.
00:21
For a, we were asked for the probability that no student arrives within 15 minutes of the previous student.
00:29
So let's say x is the number of students who arrive in 15 minutes.
00:50
We are looking for the probability that x is equal to 0.
00:57
Now, since this is a poisson process, we can say that x is a poisson random variable with a rate parameter, or we'll say with lambda t, equal to 12 times one quarter.
01:22
So 15 minutes is a quarter hour.
01:25
So we have lambda times t is equal to 3.
01:30
Our mean arrival rate is 3.
01:34
Every 15 minutes, which is the period of interest for part a.
01:41
Now the probability mass function for a poisson random variable is given by this formula.
02:01
It's e to the negative lambda t times lambda t to the exponent x over x factorial, and x takes on any non -negative integer.
02:24
So now for the probability that x is equal to 0, e to the negative 3 times 3 to the exponent 0 over 0 factorial.
02:39
And this comes out to 0 .0498.
02:52
And then for part b, we were asked for the expectation and the variance of the waiting time in minutes between the 3rd and 10th students.
03:04
Now, since the poisson process is memoryless, the waiting time between the 3rd and 10th students is the same as the waiting time for the first seven students, or the waiting time for any seven students in sequence.
03:21
The waiting time between student arrivals for a poisson process is exponentially distributed.
03:27
That's the waiting time for one more arrival.
03:31
And so if y is the random variable that is the waiting time for seven arrivals in sequence to occur, this is an erlang random variable, where 7 is the number of arrivals that we're waiting for...