If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

# Poisson process 2

## Video transcript

I think we now have all the tools we need to move forward so just to review a little bit of the last video we said we're trying to model out the probability distribution of how many cars might pass in an hour and the first thing we did is we we sat at that intersection and we found a pretty good expected value of our random variable in this random variable just to go back to the top we define the random variable is a number of cars that pass in an hour at a certain point on a certain road and we said that you know we measure it a bunch we set that they're a bunch of hours and we got a pretty good estimate of this and we say it's lambda and we said okay we wanted to model it as a binomial distribution so if this is a binomial distribution then this lambda would be equal to the number of trials times the probability of success per trial and so if we could view a trial as an interval of time this is the total number of successes in an hour success in hour and then so this would be success in a smaller interval in interval and this would be the probability of success in that smaller interval and in the last video we tried it out we said oh well what if we make this interval a minute and this is the probability of success per minute we'd have of maybe a reasonable description of what we're just grabbing vote if more than one car passes in a minute and they said oh let's make this per second and this is the probability of success per second but then we still have the problem more than one car could pass in a second very easily so what we wanted to do is we want to take the limit as this approaches infinity and then see what what kind of formula we get from the math gods so if we describe this as a binomial distribution with the limit as it approaches infinity we could say that the probability that X is equal to some number so the probability that our random variable is equal to I don't know three cars in a particular hour exactly three cars per tick in an hour is equal to oh we want to take the limit as it approaches infinity right the limit as n approaches infinity of n choose K all right we're going to have K moments in time right because n approaches infinity these ervil's become super SuperDuper small right so these become moments in time so we have we're going to have close to an infinite number of moments and this is the number of successful moments where cars pass if we have three moments where cause car where there where there was a success where a car passed and we had a total of three cars passed right or seven cars seven moments where it was true that a car passed and we would have total seven cars pass in the hour so just finishing up with our bundle and distribution end moments choose K successes times the probability of success what's the probability of success we said if this is so this could be you know if n is so this would be n what's P equal to P is equal to lambda divided by n right n times P is lambda so let me just write that down P is equal to lambda divided by n I just rearranged this up here right so our probability of success is lambda times n lambda times N and we're saying what's the probability that we have K successes and then what's the probability that we have a failure well it's 1 minus the probability of success and how many failures we're going to have how many moments will not have a car pass well we have total of n moments and K of them were successes so we'll have n minus K failures let's see what we can do with this so this is equal to let me rewrite it all and I'll change colors the limit is n approaches infinity let me write out this binomial coefficient that's n factorial over n minus K factorial times K factorial normally I write these the other way around but it's the same thing times C lambda to the K I'm just using my exponent properties over n to the K and then this expression right here I can actually separate out the X plus this is the same thing as 1 minus lambda over N to the N times 1 minus lambda over N to the minus K right you have the same base you could add the exponents and you would get this up here and let me simplify a little bit more let me let me swap spots with these two right there both you can kind of view them both as being in the denominator so you can change the order of division or multiplication depending how you view it so this is equal to the limit let me switch colors the limit as n approaches infinity I don't like that color of well what was let's let actually let me just rewrite what we did in the last video what is this thing right here and we showed it at the end of the last video n factorial divided by n minus K factorial it was n times n minus 1 times n minus 2 all the way to n minus k plus 1 right if this was 7 over 7 minus 2 factorial we would have 7 times 6 right and 6 is 1 more than 7 minus 2 so that's where we got that and we did that in the last video if you're getting confused and we also said that there's going to be exactly K terms here so if you counted these as 1 2 3 all the way there's going to be K terms here and so that we took care of that we just rewrote that and I said I would switch these two things around so that's divided by divided by n to the K n to the K times I'm just switching these lambda to the K over K factorial over K factorial and then what do we have here we have well I can just rewrite that this is continuing the same line 1 minus lambda over N to the N times 1 minus lambda over N to the minus K now we can take the limit so what happens when we take the load so just you know if you take the limit this is another property that just so you don't get too overwhelmed another property of limits if I take the limit as X approaches anything a of f of X times G of X that's equal to the limit as X approaches a of f of X times the limit as X approaches a of G of X right so we could take each of these limits in the product and then multiply them and then what all set so let's do that and I want to leave this stuff up here so first of all what's this limit let me write this out and let me pick a good color yellow so we have the limit the limit as n approaches infinity so this thing up here this n times n minus 1 times n minus 2 all the way down to n minus k plus 1 what's it going to look like it's going to be a polynomial right we're multiplying a bunch of well really we're multiplying a bunch of binomials we're doing it K times so the largest degree term is going to be n to the K right it's going to be n to the K a plus something times n to the K minus 1 it's going to be you know this big kind of binomial this big polynomial Kait degree polynomial and that's really all we need to know for this derivation so it's going to be n to the K plus bla bla bla bla bla bla bla a bunch of other stuff this thing when you multiply it out over we have this n to the K right so we just that's this part of it times the limit as well actually we don't have to worry this is a constant so we could actually bring this out front so we don't even have to write a limit so times lambda to the K over K factorial right there's no n here so this is a constant with respect to n times the limit as n approaches infinity of 1 minus lambda over N to the N times 1 minus lambda over N to the minus K all right I know you can barely see look read this let's see so first of all what's this limit the limit as n approaches infinity of some polynomial where it's n to the K power plus blah blah blah blah where all of these other terms have a lower degree this is the highest degree term right this is the highest degree term so you have n to the K in the numerator and you have n to the K in the denominator so the highest degrees are the same the coefficients are 1 so this limit is 1 another way you could do it you could divide the numerator in the denominator by n to the K and you would get you know this would be 1 over this would just be 1 plus 1 over you know n plus 1 over everything else would have a 1 over n in it and this would just be a 1 and if you take the limit as you approach infinity all of these other terms would be zero and you get left with one over one but either way you have the same degree in the top and the bottom and their coefficients are the same so the limit is n approaches infinity of this is one which is a nice simplification so you end up with 1 times lambda K over K factorial now what's the limit as n approaches infinity of this thing right here 1 minus lambda over N to the N well in the last video we show that it would be I'll write it right here that the limit as n approaches infinity of 1 plus a over N to the N is equal to e to the a right that's exactly what we have here but instead of an a we have a minus lambda right minus lambda so this is going to be equal to e to the minus lambda right we have a minus lambda instead of an A and then finally what's the limit as n approaches infinity let me write it a little bit neater of one this I'm just rewriting this term 1 minus lambda over N to the minus K power what happens is n approaches infinity well this this term right lambda is a constant is this approaches infinity this term is going to approach is 0 so you have 1 to the minus K 1 to any power is 1 so that term becomes 1 so we have another one there so there you have it we're we're done the probability the probability that our random variable the number of cars that pass in an hour is equal to a particular number you know it's equal to 7 cars passing an hour is equal to the limit as n approaches infinity of n choose K times times what we said it was lambda over N to the to the K successes times 1 minus lambda over N to the N minus K failures and we just show that this is equal to lambda to the K power over K factorial times e to the minus lambda and that's pretty neat because when you just see it in kind of a vacuum if you have no context for it you wouldn't guess that this is in any way related to the by Gnomeo theorem I mean it's got an e in there it's got a factorial but you know a lot of things have factorials in life so not clear that that would make it a binomial theorem but this is just the limit as you take smaller and smaller smaller intervals and the probability of success in each interval becomes smaller but as you take the limit you end up with E and if you think about it makes sense because one of our derivations of V actually came out of compound interest and we kind of did something similar there we took smaller and smaller intervals of compounding and over each interval we compounded by a much smaller number and when you took the limit you got a again and that's actually where that whole formula up here came up came from to begin with anyway just so that you know how to use this thing so let's say that I were to go out I'm the traffic engineer and I figure out that on average nine cars passed per hour nine cars pass and I want to know the probability I want to know the probability that so this is my expected value or and in a given hour on average nine cars are passing so I want to probably that that two cars pass in a given hour exactly two cars pass it's going to be equal to nine cars per hour to the toothpowder or four squared instead of the tooth power divided by 2 factorial times e to the minus 9 power so it's equal to 81 over 2 times e to the minus 9 power and let's see maybe I should just get the graphing calculator out there well I'll let you do that exercise to figure out what that is but I'll see you in the next video