00:01
So in this question, we're told that we have the arrivals of customers our pusson distributed with the rate, average arrival rate is lambda is 4 .32 per minute.
00:17
So that means that the probability that the amount of time from one customer to the next is equal to t is going to be e to the minus lambda t times 1 over land.
00:35
Lambda.
00:37
And that 1 over lambda is there to make sure that when we integrate this all up, we get one.
00:42
And this is a probability density, really, rather than a probability.
00:46
This is a probability density.
00:50
Okay.
00:51
So first of all, what's the probability that at least 11 minutes will elapse between arrivals? so that's going to be the probability that t is greater than or equal to 11.
01:02
It's going to be the integral from 11 to infinity of 1 over lambda e to the minus lambda t, which is, e to the minus 11t is when we enter in fact let's just show every step just so that we're clear what's going on here when we integrate this we get a minus one uh sorry this should be lambda not one over lambda um when we integrate this we get a minus one over lambda so we get minus e to the minus lambda t between 11 and infinity and t equals infinity this is going to be zero so we have to do e to the minus 11, sorry, 11 lambda.
01:53
So this is e to the power of minus 11 times lambda, which we said was 4 .32.
02:09
So this is 2 .3 times 10 to the minus 21.
02:15
So pretty much impossible.
02:18
It's almost certainly not going to be that long.
02:22
So now let's try the probability that at least four minutes will elapse.
02:27
So that's the probability that t is great than equal to four.
02:31
Well, this by all the same steps as before, is going to be e to the minus 4 lambda.
02:37
So this is 3 .13 times 10 to the minus 8...