00:01
Alright, so in 2018, we have an estimated 1 .632 million people living in manhattan, and 8 .4 million living in all of new york.
00:11
So we can represent that as a column vector here.
00:15
1 .632 in manhattan, and 8 .4 million in all five boroughs.
00:23
I'm actually going to figure out how many that is in just the other four boroughs, queens, brooklyn, staten island, and the bronx.
00:31
8 .4 minus 1 .632 is 6 .768.
00:39
6 .768 is the population living in the other three boroughs.
00:46
The other four boroughs, excuse me.
00:49
Now, in each year, 4 % of manhattan's population moves into manhattan.
00:53
Moves out of manhattan, and 3 % move into manhattan.
00:57
So we can represent that with a little matrix here.
01:01
We're saying 4 % moves out of manhattan, 0 .04, and 3 % move from the other boroughs into manhattan.
01:14
There's no other migration or population growth.
01:17
Means that in manhattan, oh excuse me, that should be here, 0 .04, and the rest, so that's going to be 96%.
01:29
96 % stay in manhattan, and 97 % stay in the outer boroughs.
01:38
If we apply this then to whatever, to our population vector here, which i'm just going to call manhattan and the rest, this will be, let's just make sure here, the new population in manhattan would be 96 % of the current population in manhattan, plus 3 % of the rest of the boroughs.
01:58
The population of the rest of the boroughs would be 4 % that were in manhattan, and 97 % of the rest.
02:03
So this is set up correctly, and we'd like to do these three parts.
02:09
A, calculate the distribution of the population in terms of manhattan and outside of manhattan in 2019, 2021, 2023, and 2024.
02:18
We're going to do that with matrices.
02:21
I'm going to do these transition matrices, but i do not want to do it manually.
02:25
So we'll look at this in different alpha.
02:28
Our transition matrix is 0 .96, 0 .03, 0 .04, 0 .97, and we're going to multiply that first by the 2018 population, so 1 .632 million in manhattan, and 6 .768 million in the rest of the boroughs.
02:58
So, in 2019, let's write this out here.
03:06
In 2019, we should expect this distribution here.
03:13
Let's make this a little bit narrower so you can see it.
03:20
1 .796...we'll do four decimal places in general, or three decimal places, four digits.
03:25
So 1 .7697 is going to be 770.
03:31
1 .770 million.
03:34
And then outside of manhattan, 6 .630.
03:39
6 .630.
03:42
In 2020, we want to just do this twice, so i'm going to square the matrix to go two years in the future.
03:53
We've got 1 .898.
03:57
In 2020, it's 1 .898 million in manhattan, and outside of manhattan, 6 .502.
04:04
6 .502.
04:06
All this, of course, is measured in millions.
04:10
In 2021, we expect this to be...
04:14
I'm going to stop writing it down, but still read it out.
04:18
2 .017 million in manhattan, 6 .383 million in the rest of the boroughs in 2021.
04:28
In 2022, that's four years after 2018, there should be 2 .218 million in manhattan, 6 .272 million outside of manhattan, and the rest of new york city.
04:38
And then in 2023, five years after, so there's one more after this, we'll go, oh, excuse me, 2 .231 million in manhattan, 6 .169 million outside of manhattan.
04:55
And finally, in 2024, six years after 2018, so we're raising it to the power of six, there should be 2 .327 million in manhattan, and 6 .073 million in the other four boroughs of new york city.
05:16
Now, i'd like to predict the long -term new york city population distribution.
05:22
One way we can do this in a relatively straightforward way is just raise this to a very high power.
05:27
So what's this going to happen after a thousand years? we expect to be around 3 .6 and 4 .8.
05:35
To do that more precisely, we can find the eigenvectors of this matrix...