[Script Info]
Title: 
[Events]
Format: Layer, Start, End, Style, Name, MarginL, MarginR, MarginV, Effect, Text
Dialogue: 0,0:00:00.86,0:00:03.54,Default,,0000,0000,0000,,Let's say you're some type of\Ntraffic engineer and what
Dialogue: 0,0:00:03.54,0:00:06.81,Default,,0000,0000,0000,,you're trying to figure out is,\Nhow many cars pass by a certain
Dialogue: 0,0:00:06.81,0:00:08.32,Default,,0000,0000,0000,,point on the street at\Nany given point in time?
Dialogue: 0,0:00:08.32,0:00:10.21,Default,,0000,0000,0000,,And you want to figure out\Nthe probabilities that a
Dialogue: 0,0:00:10.21,0:00:14.01,Default,,0000,0000,0000,,hundred cars pass or 5\Ncars pass in a given hour.
Dialogue: 0,0:00:14.01,0:00:15.81,Default,,0000,0000,0000,,So a good place to start is\Njust to define a random
Dialogue: 0,0:00:15.81,0:00:20.53,Default,,0000,0000,0000,,variable that essentially\Nrepresents what you care about.
Dialogue: 0,0:00:20.53,0:00:27.35,Default,,0000,0000,0000,,So let's say the number of cars\Nthat pass in some amount of
Dialogue: 0,0:00:27.35,0:00:30.41,Default,,0000,0000,0000,,time, let's say, in an hour.
Dialogue: 0,0:00:31.71,0:00:34.51,Default,,0000,0000,0000,,And your goal is to figure out\Nthe probability distribution of
Dialogue: 0,0:00:34.51,0:00:37.05,Default,,0000,0000,0000,,this random variable and then\Nonce you know the probability
Dialogue: 0,0:00:37.05,0:00:39.45,Default,,0000,0000,0000,,distribution then you can\Nfigure out what's the
Dialogue: 0,0:00:39.45,0:00:41.79,Default,,0000,0000,0000,,probability that 100 cars pass\Nin an hour or the probability
Dialogue: 0,0:00:41.79,0:00:45.89,Default,,0000,0000,0000,,that no cars pass in an hour\Nand you'd be unstoppable.
Dialogue: 0,0:00:45.89,0:00:48.29,Default,,0000,0000,0000,,And just a little aside, just\Nto move forward with this
Dialogue: 0,0:00:48.29,0:00:50.54,Default,,0000,0000,0000,,video, there's two assumptions\Nwe need to make because
Dialogue: 0,0:00:50.54,0:00:52.24,Default,,0000,0000,0000,,we're going to study the\NPoisson distribution.
Dialogue: 0,0:00:52.24,0:00:54.11,Default,,0000,0000,0000,,And in order to study it's\Nthere's two assumptions
Dialogue: 0,0:00:54.11,0:00:54.63,Default,,0000,0000,0000,,we have to make:
Dialogue: 0,0:00:54.63,0:00:58.77,Default,,0000,0000,0000,,That any hour at this point\Non the street is no different
Dialogue: 0,0:00:58.77,0:00:59.65,Default,,0000,0000,0000,,than any other hour.
Dialogue: 0,0:00:59.65,0:01:01.34,Default,,0000,0000,0000,,And we know that that's\Nprobably false.
Dialogue: 0,0:01:01.34,0:01:03.75,Default,,0000,0000,0000,,During rush hour in a real\Nsituation you probably
Dialogue: 0,0:01:03.75,0:01:06.64,Default,,0000,0000,0000,,would have more cars than\Nat another rush hour.
Dialogue: 0,0:01:06.64,0:01:08.64,Default,,0000,0000,0000,,And you know, if you wanted to\Nbe more realistic maybe we do
Dialogue: 0,0:01:08.64,0:01:12.37,Default,,0000,0000,0000,,it in the day because in a day\Nany period of time--
Dialogue: 0,0:01:12.37,0:01:12.75,Default,,0000,0000,0000,,actually, no.
Dialogue: 0,0:01:12.75,0:01:14.12,Default,,0000,0000,0000,,I shouldn't do a day.
Dialogue: 0,0:01:14.12,0:01:17.75,Default,,0000,0000,0000,,We have to assume that every\Nhour is completely just like
Dialogue: 0,0:01:17.75,0:01:19.65,Default,,0000,0000,0000,,any other hour and actually,\Neven within the hour there's
Dialogue: 0,0:01:19.65,0:01:22.99,Default,,0000,0000,0000,,really no differentiation from\None second to the other in
Dialogue: 0,0:01:22.99,0:01:25.82,Default,,0000,0000,0000,,terms of the probabilities\Nthat a car arrives.
Dialogue: 0,0:01:25.82,0:01:27.95,Default,,0000,0000,0000,,That's a little bit of a\Nsimplifying assumption that
Dialogue: 0,0:01:27.95,0:01:29.95,Default,,0000,0000,0000,,might not truly apply to\Ntraffic, but I think we
Dialogue: 0,0:01:29.95,0:01:32.27,Default,,0000,0000,0000,,can make that assumption.
Dialogue: 0,0:01:32.27,0:01:34.16,Default,,0000,0000,0000,,And then the other assumption\Nwe need to make is that if a
Dialogue: 0,0:01:34.16,0:01:36.69,Default,,0000,0000,0000,,bunch of cars pass in one hour\Nthat doesn't mean that fewer
Dialogue: 0,0:01:36.69,0:01:37.82,Default,,0000,0000,0000,,cars will pass in the next.
Dialogue: 0,0:01:37.82,0:01:40.63,Default,,0000,0000,0000,,That in no way does the number\Nof cars that pass in one period
Dialogue: 0,0:01:40.63,0:01:44.86,Default,,0000,0000,0000,,affect or correlate or somehow\Ninfluence the number of cars
Dialogue: 0,0:01:44.86,0:01:45.38,Default,,0000,0000,0000,,that pass in the next.
Dialogue: 0,0:01:45.38,0:01:47.37,Default,,0000,0000,0000,,That they're really\Nindependent.
Dialogue: 0,0:01:47.37,0:01:50.67,Default,,0000,0000,0000,,Given that, we can then at\Nleast try using the skills
Dialogue: 0,0:01:50.67,0:01:53.48,Default,,0000,0000,0000,,we have to model out some\Ntype of a distribution.
Dialogue: 0,0:01:53.48,0:01:55.77,Default,,0000,0000,0000,,The first thing you do and I'd\Nrecommend doing this for any
Dialogue: 0,0:01:55.77,0:01:59.09,Default,,0000,0000,0000,,distribution is maybe we\Ncan estimate the mean.
Dialogue: 0,0:01:59.09,0:02:03.04,Default,,0000,0000,0000,,Let's sit out on that curb and\Nmeasure what this variable is
Dialogue: 0,0:02:03.04,0:02:05.17,Default,,0000,0000,0000,,over a bunch of hours and then\Naverage it up, and that's going
Dialogue: 0,0:02:05.17,0:02:08.89,Default,,0000,0000,0000,,to be a pretty good estimator\Nfor the actual mean
Dialogue: 0,0:02:08.89,0:02:09.88,Default,,0000,0000,0000,,of our population.
Dialogue: 0,0:02:09.88,0:02:12.27,Default,,0000,0000,0000,,Or, since it's a random\Nvariable, the expected value
Dialogue: 0,0:02:12.27,0:02:13.01,Default,,0000,0000,0000,,of this random variable.
Dialogue: 0,0:02:13.01,0:02:16.66,Default,,0000,0000,0000,,Let's say you do that and you\Nget your best estimate of the
Dialogue: 0,0:02:16.66,0:02:22.27,Default,,0000,0000,0000,,expected value of this random\Nvariable is-- I'll use
Dialogue: 0,0:02:22.27,0:02:24.85,Default,,0000,0000,0000,,the letter lambda.
Dialogue: 0,0:02:24.85,0:02:27.38,Default,,0000,0000,0000,,You know, this could\Nbe 9 cars per hour.
Dialogue: 0,0:02:27.38,0:02:30.19,Default,,0000,0000,0000,,You sat out there-- it could\Nbe 9.3 cars per hour.
Dialogue: 0,0:02:30.19,0:02:32.67,Default,,0000,0000,0000,,You sat out there over hundreds\Nof hours and you just counted
Dialogue: 0,0:02:32.67,0:02:34.59,Default,,0000,0000,0000,,the number of cars each hour\Nand you averaged them all up.
Dialogue: 0,0:02:34.59,0:02:37.25,Default,,0000,0000,0000,,You said, on average, there are\N9.3 cars per hour and you feel
Dialogue: 0,0:02:37.25,0:02:38.68,Default,,0000,0000,0000,,that's a pretty good estimate.
Dialogue: 0,0:02:38.68,0:02:40.08,Default,,0000,0000,0000,,So that's what you have there.
Dialogue: 0,0:02:40.08,0:02:42.00,Default,,0000,0000,0000,,And let's see what we could do.
Dialogue: 0,0:02:42.00,0:02:45.56,Default,,0000,0000,0000,,We know the binomial\Ndistribution.
Dialogue: 0,0:02:45.56,0:02:50.65,Default,,0000,0000,0000,,The binomial distribution tells\Nus that the expected value of a
Dialogue: 0,0:02:50.65,0:02:55.22,Default,,0000,0000,0000,,random variable is equal to the\Nnumber of trials that that
Dialogue: 0,0:02:55.22,0:02:57.46,Default,,0000,0000,0000,,random variable's kind\Nof composed of, right?
Dialogue: 0,0:02:57.46,0:02:59.49,Default,,0000,0000,0000,,Before, in the previous videos\Nwe were counting the number
Dialogue: 0,0:02:59.49,0:03:00.50,Default,,0000,0000,0000,,of heads in a coin toss.
Dialogue: 0,0:03:00.50,0:03:03.07,Default,,0000,0000,0000,,So this would be the number\Nof coin tosses, times the
Dialogue: 0,0:03:03.07,0:03:07.29,Default,,0000,0000,0000,,probability of success\Nover each toss.
Dialogue: 0,0:03:07.29,0:03:09.00,Default,,0000,0000,0000,,This is what we did with\Nthe binomial distribution.
Dialogue: 0,0:03:09.00,0:03:11.67,Default,,0000,0000,0000,,So maybe we can model\Nour traffic situation
Dialogue: 0,0:03:11.67,0:03:12.78,Default,,0000,0000,0000,,something similar.
Dialogue: 0,0:03:12.78,0:03:15.40,Default,,0000,0000,0000,,This is the number of cars\Nthat pass in an hour.
Dialogue: 0,0:03:15.40,0:03:22.80,Default,,0000,0000,0000,,So maybe we could say lambda\Ncars per hour is equal
Dialogue: 0,0:03:22.80,0:03:24.33,Default,,0000,0000,0000,,to-- I don't know.
Dialogue: 0,0:03:26.85,0:03:29.88,Default,,0000,0000,0000,,Let's make each experiment or\Neach toss of the coin equal to
Dialogue: 0,0:03:29.88,0:03:31.78,Default,,0000,0000,0000,,whether a car passes\Nin a given minute.
Dialogue: 0,0:03:31.78,0:03:37.98,Default,,0000,0000,0000,,So there are 60 minutes\Nper hour, so there
Dialogue: 0,0:03:37.98,0:03:40.87,Default,,0000,0000,0000,,would be 60 trials.
Dialogue: 0,0:03:40.87,0:03:43.19,Default,,0000,0000,0000,,And then, the probability that\Nwe have success in each of
Dialogue: 0,0:03:43.19,0:03:46.99,Default,,0000,0000,0000,,those trials, if we modeled\Nthis as a binomial distribution
Dialogue: 0,0:03:46.99,0:03:54.45,Default,,0000,0000,0000,,would be lambda over\N60 cars per minute.
Dialogue: 0,0:03:54.45,0:03:55.66,Default,,0000,0000,0000,,And this would be\Na probability.
Dialogue: 0,0:03:55.66,0:03:58.64,Default,,0000,0000,0000,,This would be n, and this would\Nbe the probability, if we said
Dialogue: 0,0:03:58.64,0:04:00.27,Default,,0000,0000,0000,,that this is a binomial\Ndistribution.
Dialogue: 0,0:04:00.27,0:04:04.03,Default,,0000,0000,0000,,And this probably wouldn't be\Nthat bad of an approximation.
Dialogue: 0,0:04:04.03,0:04:06.13,Default,,0000,0000,0000,,If you actually then said,\Noh, this is a binomial
Dialogue: 0,0:04:06.13,0:04:10.38,Default,,0000,0000,0000,,distribution, so the\Nprobability that our random
Dialogue: 0,0:04:10.38,0:04:12.94,Default,,0000,0000,0000,,variable equals some\Ngiven value, k.
Dialogue: 0,0:04:12.94,0:04:16.17,Default,,0000,0000,0000,,You know, the probability that\N3 cars, exactly 3 cars pass in
Dialogue: 0,0:04:16.17,0:04:19.75,Default,,0000,0000,0000,,an given hour, we would\Nthen be equal to n.
Dialogue: 0,0:04:19.75,0:04:21.89,Default,,0000,0000,0000,,So n would be 60.
Dialogue: 0,0:04:21.89,0:04:26.01,Default,,0000,0000,0000,,Choose k, and you know,\NI have 3 cars times the
Dialogue: 0,0:04:26.01,0:04:27.19,Default,,0000,0000,0000,,probability of success.
Dialogue: 0,0:04:27.19,0:04:29.57,Default,,0000,0000,0000,,So the probability that a\Ncar passes in any minute.
Dialogue: 0,0:04:29.57,0:04:34.77,Default,,0000,0000,0000,,So it'd be lambda over\N60 to the number of
Dialogue: 0,0:04:34.77,0:04:35.98,Default,,0000,0000,0000,,successes we need.
Dialogue: 0,0:04:35.98,0:04:41.66,Default,,0000,0000,0000,,So to the kth power, times the\Nprobability of no success or
Dialogue: 0,0:04:41.66,0:04:46.56,Default,,0000,0000,0000,,that no cars pass,\Nto the n minus k.
Dialogue: 0,0:04:46.56,0:04:50.23,Default,,0000,0000,0000,,If we have k successes we have\Nto have 60 minus k failures.
Dialogue: 0,0:04:50.23,0:04:52.95,Default,,0000,0000,0000,,There are 60 minus k minutes\Nwhere no car passed.
Dialogue: 0,0:04:52.95,0:04:55.27,Default,,0000,0000,0000,,This actually wouldn't be that\Nbad of an approximation where
Dialogue: 0,0:04:55.27,0:04:57.25,Default,,0000,0000,0000,,you have 60 intervals and you\Nsay this is a binomial
Dialogue: 0,0:04:57.25,0:04:58.56,Default,,0000,0000,0000,,distribution.
Dialogue: 0,0:04:58.56,0:05:00.31,Default,,0000,0000,0000,,And you'd probably get\Nreasonable results.
Dialogue: 0,0:05:00.31,0:05:02.60,Default,,0000,0000,0000,,But there's a core issue here.
Dialogue: 0,0:05:02.60,0:05:06.58,Default,,0000,0000,0000,,In this model where we model it\Nas a binomial distribution,
Dialogue: 0,0:05:06.58,0:05:09.98,Default,,0000,0000,0000,,what happens if more than\None car passes in an hour?
Dialogue: 0,0:05:09.98,0:05:11.63,Default,,0000,0000,0000,,Or more than one car\Npasses in a minute?
Dialogue: 0,0:05:11.63,0:05:14.27,Default,,0000,0000,0000,,The way we have it right now\Nwe call it a success if one
Dialogue: 0,0:05:14.27,0:05:15.32,Default,,0000,0000,0000,,car passes in a minute.
Dialogue: 0,0:05:15.32,0:05:18.79,Default,,0000,0000,0000,,And if you're kind of counting\Nit counts as one success, even
Dialogue: 0,0:05:18.79,0:05:21.19,Default,,0000,0000,0000,,if 5 cars pass in that minute.
Dialogue: 0,0:05:21.19,0:05:23.39,Default,,0000,0000,0000,,So you say, oh, OK Sal, I\Nknow the solution there.
Dialogue: 0,0:05:23.39,0:05:26.04,Default,,0000,0000,0000,,I just have to get\Nmore granular.
Dialogue: 0,0:05:26.04,0:05:28.87,Default,,0000,0000,0000,,Instead of dividing it\Ninto minutes why don't I
Dialogue: 0,0:05:28.87,0:05:31.05,Default,,0000,0000,0000,,divide it into seconds?
Dialogue: 0,0:05:31.05,0:05:36.21,Default,,0000,0000,0000,,So the probability that I have\Nk successes-- instead of 60
Dialogue: 0,0:05:36.21,0:05:39.82,Default,,0000,0000,0000,,intervals I'll do\N3,600 intervals.
Dialogue: 0,0:05:39.82,0:05:43.17,Default,,0000,0000,0000,,So the probability of k\Nsuccessful seconds, so a second
Dialogue: 0,0:05:43.17,0:05:48.61,Default,,0000,0000,0000,,where a car is passing at that\Nmoment out of 3,600 seconds.
Dialogue: 0,0:05:48.61,0:05:52.19,Default,,0000,0000,0000,,So that's 3,600 choose k, times\Nthe probability that a car
Dialogue: 0,0:05:52.19,0:05:55.21,Default,,0000,0000,0000,,passes in any given second.
Dialogue: 0,0:05:55.21,0:05:57.93,Default,,0000,0000,0000,,That's the expected number of\Ncars in an hour divided by
Dialogue: 0,0:05:57.93,0:06:00.43,Default,,0000,0000,0000,,number seconds in an hour.
Dialogue: 0,0:06:00.43,0:06:01.40,Default,,0000,0000,0000,,We're going to\Nhave k successes.
Dialogue: 0,0:06:03.99,0:06:06.27,Default,,0000,0000,0000,,And these are the failures,\Nthe probability of a failure
Dialogue: 0,0:06:06.27,0:06:12.05,Default,,0000,0000,0000,,and you're going to have\N3,600 minus k failures.
Dialogue: 0,0:06:12.05,0:06:13.91,Default,,0000,0000,0000,,And this would be even a\Nbetter approximation.
Dialogue: 0,0:06:13.91,0:06:16.77,Default,,0000,0000,0000,,This actually would not be so\Nbad, but still, you have this
Dialogue: 0,0:06:16.77,0:06:19.10,Default,,0000,0000,0000,,situation where 2 cars\Ncan come within a half a
Dialogue: 0,0:06:19.10,0:06:19.98,Default,,0000,0000,0000,,second of each other.
Dialogue: 0,0:06:19.98,0:06:21.91,Default,,0000,0000,0000,,And you say, oh, OK Sal,\NI see the pattern here.
Dialogue: 0,0:06:21.91,0:06:23.65,Default,,0000,0000,0000,,We just have to get more\Nand more granular.
Dialogue: 0,0:06:23.65,0:06:26.17,Default,,0000,0000,0000,,We have to just make\Nthis number larger and
Dialogue: 0,0:06:26.17,0:06:27.40,Default,,0000,0000,0000,,larger and larger.
Dialogue: 0,0:06:27.40,0:06:28.95,Default,,0000,0000,0000,,And your intuition is correct.
Dialogue: 0,0:06:28.95,0:06:31.34,Default,,0000,0000,0000,,And if you do that you'll\Nend up getting the
Dialogue: 0,0:06:31.34,0:06:33.86,Default,,0000,0000,0000,,Poisson distribution.
Dialogue: 0,0:06:33.86,0:06:35.62,Default,,0000,0000,0000,,And this is really interesting\Nbecause a lot of times people
Dialogue: 0,0:06:35.62,0:06:38.60,Default,,0000,0000,0000,,give you the formula for the\NPoisson distribution and you
Dialogue: 0,0:06:38.60,0:06:40.42,Default,,0000,0000,0000,,can kind of just plug in\Nthe numbers and use it.
Dialogue: 0,0:06:40.42,0:06:43.25,Default,,0000,0000,0000,,But it's neat to know that it\Nreally is just the binomial
Dialogue: 0,0:06:43.25,0:06:45.79,Default,,0000,0000,0000,,distribution and the binomial\Ndistribution really did come
Dialogue: 0,0:06:45.79,0:06:48.59,Default,,0000,0000,0000,,from kind of the common\Nsense of flipping coins.
Dialogue: 0,0:06:48.59,0:06:50.50,Default,,0000,0000,0000,,That's where everything\Nis coming from.
Dialogue: 0,0:06:50.50,0:06:53.71,Default,,0000,0000,0000,,But before we kind of prove\Nthat if we take the limit
Dialogue: 0,0:06:53.71,0:06:55.67,Default,,0000,0000,0000,,as-- let me change colors.
Dialogue: 0,0:06:55.67,0:06:58.47,Default,,0000,0000,0000,,Before we proved that as we\Ntake the limit as this number
Dialogue: 0,0:06:58.47,0:07:01.27,Default,,0000,0000,0000,,right here, the number of\Nintervals approaches infinity
Dialogue: 0,0:07:01.27,0:07:04.07,Default,,0000,0000,0000,,that this becomes the\NPoisson distribution.
Dialogue: 0,0:07:04.07,0:07:07.29,Default,,0000,0000,0000,,I'm going to make sure we have\Na couple of mathematical
Dialogue: 0,0:07:07.29,0:07:09.15,Default,,0000,0000,0000,,tools in our belt.
Dialogue: 0,0:07:09.15,0:07:12.76,Default,,0000,0000,0000,,So the first is something that\Nyou're probably reasonably
Dialogue: 0,0:07:12.76,0:07:15.86,Default,,0000,0000,0000,,familiar with by now, but I\Njust want to make sure that the
Dialogue: 0,0:07:15.86,0:07:25.68,Default,,0000,0000,0000,,limit as x approaches infinity\Nof 1 plus a/x to the x power is
Dialogue: 0,0:07:25.68,0:07:31.02,Default,,0000,0000,0000,,equal to e to the\Nax-- no sorry.
Dialogue: 0,0:07:31.02,0:07:38.02,Default,,0000,0000,0000,,Is equal to e to the a and now\Njust to prove this to you,
Dialogue: 0,0:07:38.02,0:07:39.26,Default,,0000,0000,0000,,let's make a little\Nsubstitution here.
Dialogue: 0,0:07:39.26,0:07:43.64,Default,,0000,0000,0000,,Let's say that n is equal\Nto-- let me say 1 over
Dialogue: 0,0:07:43.64,0:07:47.88,Default,,0000,0000,0000,,n is equal to a over x.
Dialogue: 0,0:07:47.88,0:07:52.89,Default,,0000,0000,0000,,And then what would be\Nx would equal to na.
Dialogue: 0,0:07:52.89,0:07:55.29,Default,,0000,0000,0000,,x times 1 is equal\Nto n times a.
Dialogue: 0,0:07:55.29,0:08:00.05,Default,,0000,0000,0000,,And so the limit as x\Napproaches infinity,
Dialogue: 0,0:08:00.05,0:08:02.04,Default,,0000,0000,0000,,what does a approach?
Dialogue: 0,0:08:02.04,0:08:02.88,Default,,0000,0000,0000,,a is-- sorry.
Dialogue: 0,0:08:02.88,0:08:04.92,Default,,0000,0000,0000,,As x approaches infinity\Nwhat does n approach?
Dialogue: 0,0:08:04.92,0:08:07.35,Default,,0000,0000,0000,,Well n is x divided by a.
Dialogue: 0,0:08:07.35,0:08:08.71,Default,,0000,0000,0000,,So n would also\Napproach infinity.
Dialogue: 0,0:08:08.71,0:08:10.81,Default,,0000,0000,0000,,So this thing would be the same\Nthing as just making our
Dialogue: 0,0:08:10.81,0:08:16.46,Default,,0000,0000,0000,,substitution the limit as n\Napproaches infinity of 1
Dialogue: 0,0:08:16.46,0:08:21.39,Default,,0000,0000,0000,,plus-- a/x, I made the\Nsubstitution as 1/n.
Dialogue: 0,0:08:21.39,0:08:26.72,Default,,0000,0000,0000,,And x is, by this\Nsubstitution, n times a.
Dialogue: 0,0:08:26.72,0:08:30.50,Default,,0000,0000,0000,,And this is going to be the\Nsame thing as the limit as n
Dialogue: 0,0:08:30.50,0:08:36.09,Default,,0000,0000,0000,,approaches infinity of 1 plus\N1/n to the n, all
Dialogue: 0,0:08:36.09,0:08:39.39,Default,,0000,0000,0000,,of that to the a.
Dialogue: 0,0:08:39.39,0:08:41.76,Default,,0000,0000,0000,,And since there's no n out here\Nwe could just take the limit
Dialogue: 0,0:08:41.76,0:08:43.45,Default,,0000,0000,0000,,of this and then take\Nthat to the a power.
Dialogue: 0,0:08:43.45,0:08:47.69,Default,,0000,0000,0000,,So that's going to be equal to\Nthe limit as n approaches
Dialogue: 0,0:08:47.69,0:08:52.60,Default,,0000,0000,0000,,infinity of 1 plus 1/n to the\Nnth power, all of
Dialogue: 0,0:08:52.60,0:08:53.78,Default,,0000,0000,0000,,that to the a.
Dialogue: 0,0:08:53.78,0:08:58.04,Default,,0000,0000,0000,,And this is our definition, or\None of the ways to get to e if
Dialogue: 0,0:08:58.04,0:09:00.82,Default,,0000,0000,0000,,you'd watch the videos on\Ncompound interest and all that.
Dialogue: 0,0:09:00.82,0:09:01.88,Default,,0000,0000,0000,,This is how we got to e.
Dialogue: 0,0:09:01.88,0:09:03.46,Default,,0000,0000,0000,,And if you tried it out on your\Ncalculator, just try larger
Dialogue: 0,0:09:03.46,0:09:07.26,Default,,0000,0000,0000,,and larger n's here\Nand you'll get e.
Dialogue: 0,0:09:07.26,0:09:12.01,Default,,0000,0000,0000,,This inner part is equal to e,\Nand we raised it to the a
Dialogue: 0,0:09:12.01,0:09:14.06,Default,,0000,0000,0000,,power, so it's equal\Nto e to the a.
Dialogue: 0,0:09:14.06,0:09:16.24,Default,,0000,0000,0000,,So hopefully you pretty\Nsatisfied that this limit
Dialogue: 0,0:09:16.24,0:09:17.86,Default,,0000,0000,0000,,is equal to e to the a.
Dialogue: 0,0:09:17.86,0:09:19.86,Default,,0000,0000,0000,,And then one other tool kit I\Nwant in our belt, and I'll
Dialogue: 0,0:09:19.86,0:09:22.34,Default,,0000,0000,0000,,probably actually do the\Nproof in the next video.
Dialogue: 0,0:09:22.34,0:09:32.95,Default,,0000,0000,0000,,The other tool kit is to\Nrecognize that x factorial over
Dialogue: 0,0:09:32.95,0:09:42.86,Default,,0000,0000,0000,,x minus k factorial is equal to\Nx times x minus 1 times x
Dialogue: 0,0:09:42.86,0:09:50.03,Default,,0000,0000,0000,,minus 2, all the way down\Nto times x minus k plus 1.
Dialogue: 0,0:09:50.03,0:09:51.88,Default,,0000,0000,0000,,And we've done this a lot of\Ntimes, but this is the most
Dialogue: 0,0:09:51.88,0:09:53.06,Default,,0000,0000,0000,,abstract we've ever written it.
Dialogue: 0,0:09:53.06,0:09:55.58,Default,,0000,0000,0000,,I can give you a couple of--\Nand just so you know, they'll
Dialogue: 0,0:09:55.58,0:09:57.33,Default,,0000,0000,0000,,be exactly k terms here.
Dialogue: 0,0:09:57.33,0:10:01.70,Default,,0000,0000,0000,,1, 2, 3-- So first term, second\Nterm, third term, all the
Dialogue: 0,0:10:01.70,0:10:04.31,Default,,0000,0000,0000,,way, and this the kth term.
Dialogue: 0,0:10:04.31,0:10:07.21,Default,,0000,0000,0000,,And this is important to\Nour derivation of the
Dialogue: 0,0:10:07.21,0:10:09.16,Default,,0000,0000,0000,,Poisson distribution.
Dialogue: 0,0:10:09.16,0:10:13.87,Default,,0000,0000,0000,,But just to make this in real\Nnumbers, if I had 7 factorial
Dialogue: 0,0:10:13.87,0:10:20.11,Default,,0000,0000,0000,,over 7 minus 2 factorial,\Nthat's equal to 7 times 6
Dialogue: 0,0:10:20.11,0:10:24.07,Default,,0000,0000,0000,,times 5 times 4 times\N3 times 3 times 1.
Dialogue: 0,0:10:24.07,0:10:27.36,Default,,0000,0000,0000,,Over 2 times-- no sorry.
Dialogue: 0,0:10:27.36,0:10:28.94,Default,,0000,0000,0000,,7 minus 2, this is 5.
Dialogue: 0,0:10:28.94,0:10:33.50,Default,,0000,0000,0000,,So it's over 5 times 4\Ntimes 3 times 2 times 1.
Dialogue: 0,0:10:33.50,0:10:37.19,Default,,0000,0000,0000,,These cancel out and you\Njust have 7 times 6.
Dialogue: 0,0:10:37.19,0:10:40.99,Default,,0000,0000,0000,,And so it's 7 and then\Nthe last term is 7 minus
Dialogue: 0,0:10:40.99,0:10:43.04,Default,,0000,0000,0000,,2 plus 1, which is 6.
Dialogue: 0,0:10:47.56,0:10:51.29,Default,,0000,0000,0000,,In this example, k was 2 and\Nyou had exactly 2 terms.
Dialogue: 0,0:10:51.29,0:10:53.23,Default,,0000,0000,0000,,So once we know those two\Nthings we're now ready
Dialogue: 0,0:10:53.23,0:10:55.71,Default,,0000,0000,0000,,to derive the Poisson\Ndistribution and I'll do
Dialogue: 0,0:10:55.71,0:10:58.42,Default,,0000,0000,0000,,that in the next video.
Dialogue: 0,0:10:58.42,0:10:59.98,Default,,0000,0000,0000,,See you soon.