00:01
Hello, so part one, we know that if t is the time of the first arrival in a poisson process with rate lambda is equal to two, then t follows an exponential distribution with parameter lambda, and therefore the pdf of t is going to be f sub t of t is equal to lambda times e to the negative lambda t, which is going to be equal to 2 times e to the negative 2t, where t is greater than or equal to 0.
00:35
Okay, and then the second arrival in the poisson process will occur at time here, t plus t, where t is another variable, another exponential variable with rate lambda, lambda and is independent of t.
00:53
So the sum of the two exponential random variables with the same rate is a gamma distribution.
01:00
And we get then that it follows a gamma distribution with shape parameter 2 and rate lambda equals 2.
01:10
And this gives us that f sub s of s is going to be equal to lambda squared times se e to the negative lambda s, which is going to be equal to 4 times s e to the negative 2s, where s is greater than or equal to 0.
01:30
Hello.
01:30
So probability here, i'm using the cdf of the exponential distribution...