Poisson Process 1 Introduction to Poisson Processes and the Poisson Distribution.
Poisson Process 1
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- Let's say you're some type of traffic engineer and what
- you're trying to figure out is, how many cars pass by a certain
- point on the street at any given point in time?
- And you want to figure out the probabilities that a
- hundred cars pass or 5 cars pass in a given hour.
- So a good place to start is just to define a random
- variable that essentially represents what you care about.
- So let's say the number of cars that pass in some amount of
- time, let's say, in an hour.
- And your goal is to figure out the probability distribution of
- this random variable and then once you know the probability
- distribution then you can figure out what's the
- probability that 100 cars pass in an hour or the probability
- that no cars pass in an hour and you'd be unstoppable.
- And just a little aside, just to move forward with this
- video, there's two assumptions we need to make because
- we're going to study the Poisson distribution.
- And in order to study it's there's two assumptions
- we have to make:
- That any hour at this point on the street is no different
- than any other hour.
- And we know that that's probably false.
- During rush hour in a real situation you probably
- would have more cars than at another rush hour.
- And you know, if you wanted to be more realistic maybe we do
- it in the day because in a day any period of time--
- actually, no.
- I shouldn't do a day.
- We have to assume that every hour is completely just like
- any other hour and actually, even within the hour there's
- really no differentiation from one second to the other in
- terms of the probabilities that a car arrives.
- That's a little bit of a simplifying assumption that
- might not truly apply to traffic, but I think we
- can make that assumption.
- And then the other assumption we need to make is that if a
- bunch of cars pass in one hour that doesn't mean that fewer
- cars will pass in the next.
- That in no way does the number of cars that pass in one period
- affect or correlate or somehow influence the number of cars
- that pass in the next.
- That they're really independent.
- Given that, we can then at least try using the skills
- we have to model out some type of a distribution.
- The first thing you do and I'd recommend doing this for any
- distribution is maybe we can estimate the mean.
- Let's sit out on that curb and measure what this variable is
- over a bunch of hours and then average it up, and that's going
- to be a pretty good estimator for the actual mean
- of our population.
- Or, since it's a random variable, the expected value
- of this random variable.
- Let's say you do that and you get your best estimate of the
- expected value of this random variable is-- I'll use
- the letter lambda.
- You know, this could be 9 cars per hour.
- You sat out there-- it could be 9.3 cars per hour.
- You sat out there over hundreds of hours and you just counted
- the number of cars each hour and you averaged them all up.
- You said, on average, there are 9.3 cars per hour and you feel
- that's a pretty good estimate.
- So that's what you have there.
- And let's see what we could do.
- We know the binomial distribution.
- The binomial distribution tells us that the expected value of a
- random variable is equal to the number of trials that that
- random variable's kind of composed of, right?
- Before, in the previous videos we were counting the number
- of heads in a coin toss.
- So this would be the number of coin tosses, times the
- probability of success over each toss.
- This is what we did with the binomial distribution.
- So maybe we can model our traffic situation
- something similar.
- This is the number of cars that pass in an hour.
- So maybe we could say lambda cars per hour is equal
- to-- I don't know.
- Let's make each experiment or each toss of the coin equal to
- whether a car passes in a given minute.
- So there are 60 minutes per hour, so there
- would be 60 trials.
- And then, the probability that we have success in each of
- those trials, if we modeled this as a binomial distribution
- would be lambda over 60 cars per minute.
- And this would be a probability.
- This would be n, and this would be the probability, if we said
- that this is a binomial distribution.
- And this probably wouldn't be that bad of an approximation.
- If you actually then said, oh, this is a binomial
- distribution, so the probability that our random
- variable equals some given value, k.
- You know, the probability that 3 cars, exactly 3 cars pass in
- an given hour, we would then be equal to n.
- So n would be 60.
- Choose k, and you know, I have 3 cars times the
- probability of success.
- So the probability that a car passes in any minute.
- So it'd be lambda over 60 to the number of
- successes we need.
- So to the kth power, times the probability of no success or
- that no cars pass, to the n minus k.
- If we have k successes we have to have 60 minus k failures.
- There are 60 minus k minutes where no car passed.
- This actually wouldn't be that bad of an approximation where
- you have 60 intervals and you say this is a binomial
- And you'd probably get reasonable results.
- But there's a core issue here.
- In this model where we model it as a binomial distribution,
- what happens if more than one car passes in an hour?
- Or more than one car passes in a minute?
- The way we have it right now we call it a success if one
- car passes in a minute.
- And if you're kind of counting it counts as one success, even
- if 5 cars pass in that minute.
- So you say, oh, OK Sal, I know the solution there.
- I just have to get more granular.
- Instead of dividing it into minutes why don't I
- divide it into seconds?
- So the probability that I have k successes-- instead of 60
- intervals I'll do 3,600 intervals.
- So the probability of k successful seconds, so a second
- where a car is passing at that moment out of 3,600 seconds.
- So that's 3,600 choose k, times the probability that a car
- passes in any given second.
- That's the expected number of cars in an hour divided by
- number seconds in an hour.
- We're going to have k successes.
- And these are the failures, the probability of a failure
- and you're going to have 3,600 minus k failures.
- And this would be even a better approximation.
- This actually would not be so bad, but still, you have this
- situation where 2 cars can come within a half a
- second of each other.
- And you say, oh, OK Sal, I see the pattern here.
- We just have to get more and more granular.
- We have to just make this number larger and
- larger and larger.
- And your intuition is correct.
- And if you do that you'll end up getting the
- Poisson distribution.
- And this is really interesting because a lot of times people
- give you the formula for the Poisson distribution and you
- can kind of just plug in the numbers and use it.
- But it's neat to know that it really is just the binomial
- distribution and the binomial distribution really did come
- from kind of the common sense of flipping coins.
- That's where everything is coming from.
- But before we kind of prove that if we take the limit
- as-- let me change colors.
- Before we proved that as we take the limit as this number
- right here, the number of intervals approaches infinity
- that this becomes the Poisson distribution.
- I'm going to make sure we have a couple of mathematical
- tools in our belt.
- So the first is something that you're probably reasonably
- familiar with by now, but I just want to make sure that the
- limit as x approaches infinity of 1 plus a/x to the x power is
- equal to e to the ax-- no sorry.
- Is equal to e to the a and now just to prove this to you,
- let's make a little substitution here.
- Let's say that n is equal to-- let me say 1 over
- n is equal to a over x.
- And then what would be x would equal to na.
- x times 1 is equal to n times a.
- And so the limit as x approaches infinity,
- what does a approach?
- a is-- sorry.
- As x approaches infinity what does n approach?
- Well n is x divided by a.
- So n would also approach infinity.
- So this thing would be the same thing as just making our
- substitution the limit as n approaches infinity of 1
- plus-- a/x, I made the substitution as 1/n.
- And x is, by this substitution, n times a.
- And this is going to be the same thing as the limit as n
- approaches infinity of 1 plus 1/n to the n, all
- of that to the a.
- And since there's no n out here we could just take the limit
- of this and then take that to the a power.
- So that's going to be equal to the limit as n approaches
- infinity of 1 plus 1/n to the nth power, all of
- that to the a.
- And this is our definition, or one of the ways to get to e if
- you'd watch the videos on compound interest and all that.
- This is how we got to e.
- And if you tried it out on your calculator, just try larger
- and larger n's here and you'll get e.
- This inner part is equal to e, and we raised it to the a
- power, so it's equal to e to the a.
- So hopefully you pretty satisfied that this limit
- is equal to e to the a.
- And then one other tool kit I want in our belt, and I'll
- probably actually do the proof in the next video.
- The other tool kit is to recognize that x factorial over
- x minus k factorial is equal to x times x minus 1 times x
- minus 2, all the way down to times x minus k plus 1.
- And we've done this a lot of times, but this is the most
- abstract we've ever written it.
- I can give you a couple of-- and just so you know, they'll
- be exactly k terms here.
- 1, 2, 3-- So first term, second term, third term, all the
- way, and this the kth term.
- And this is important to our derivation of the
- Poisson distribution.
- But just to make this in real numbers, if I had 7 factorial
- over 7 minus 2 factorial, that's equal to 7 times 6
- times 5 times 4 times 3 times 3 times 1.
- Over 2 times-- no sorry.
- 7 minus 2, this is 5.
- So it's over 5 times 4 times 3 times 2 times 1.
- These cancel out and you just have 7 times 6.
- And so it's 7 and then the last term is 7 minus
- 2 plus 1, which is 6.
- In this example, k was 2 and you had exactly 2 terms.
- So once we know those two things we're now ready
- to derive the Poisson distribution and I'll do
- that in the next video.
- See you soon.
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