00:01
So we're told that the length of phone calls follows an exponential distribution with a parameter of lambda equals 1 eighth.
00:08
And if we arrive at a phone booth immediately after somebody else, we're interested in x being the length of time that we wait, which is essentially being the length of the person's phone call ahead of us.
00:24
And so first or us, what is the probability that x is greater than five? or that the person immediately ahead of us, their phone call is greater than five minutes.
00:32
Well, theoretically, x is not bounded in the positive direction, meaning that x could go up to infinity.
00:39
And so an easier way to solve this is going to say that it's equal to 1 minus the probability that x is less than or equal to 5.
00:48
And then that allows us to do a definite integral cross the probability distribution function from 0 to 5 of 1 eighths times e to the negative 1 eighth x, which is equal to 1 .1 .5x, which is equal to 1 minus negative.
01:07
Okay, actually i'm going to run the space.
01:13
So let's just solve the integral for now on a new page.
01:26
This is going to evaluate to negative 1 over 64.
01:34
Oh wait, no, sorry, i'm doing the integral.
01:37
So this is actually going just to be equal to negative e to the negative 1 eighth x.
01:43
And you can tell that the derivative of this is going to be positive 1 eighth, e to the negative 1 of the 8th, so this is the integral evaluated from 0 to 5 gives us negative e to the negative 5 eighth minus negative e to the 0.
02:01
This is equal to 1.
02:03
And so this gives us 1 minus, see how we're reversing the negatives here, 1 minus e to the negative 5 eighths, which is equal to 0 .46, 47.
02:27
The probability that the phone call is less than five minutes.
02:30
So the probability that we wait more than five minutes is equal to 1 minus 0 .4647, which is equal to 0 .5353.
02:48
Sorry, okay, let's rewrite that...