Joint PDF of a spatial distribution

[Kalinka35]

Homework Statement

In a square town that is 1.5 miles by 1.5 miles, the places of residence are uniformly distributed (2000 per square mile) over the whole town.
Compute the joint probability density function for the spatial distribution of places of residence (f_{X, Y}).

Homework Equations

The Attempt at a Solution

I've set it up so the lower left corner of the town is at the origin.
I understand how a uniform distribution works, but the addition of the 2000 residences per square mile is really throwing me off.
I am thinking of trying to find the CDF first and then differentiating but I don't have an idea of how I would begin to find that.

 

[lanedance]

well as its a uniform distribution, then number of houses will be directly proportional to area

so i think maybe rather than writing the pdf in terms of probability, you could use the (2000 per square mile) to write number of houses as a function of x & y

then $f_{X,Y}(X=x,Y=y)dxdy$ would represent the number of houses contianed in x & x+dx, and y + y+dx

 

[Kalinka35]

Okay, so the number of houses would be given by 2000xy, if I am not mistaken.
$f_{X,Y}(X=x,Y=y)dxdy$ isn't clear to me. Wouldn't you need to multiply by 2000?

 

[lanedance]

Okay, so the number of houses would be given by 2000xy, if I am not mistaken.

what do you mean by 2000xy?

 

[Kalinka35]

Since there are 2000 houses per square mile, the total number of houses over a certain area would be 2000 time that area. The area would be given by xy. So in this case we have 2000(1.5)(1.5)=4500.
Is that what you meant?

 

[lanedance]

ok, so that sounds like the amount of houses given X<x, and Y<y

 

[lanedance]

think about how this realtes to the density function, and how the density function is related to area

 

[Kalinka35]

Well, what I've come up with is that for small dx and dy, the number of houses in that small region, R, is 2000 dxdy.
So we essentially want to find the number of residences in R over the total residences in the town. The total residences in the town is known to be 4500 so to find the residences in R would you integrate 2000dxdy? I guess that doesn't really make sense to me.

 

[lanedance]

sounds reasonable to me, think its coming together...

notice the pdf is 2000dxdy, which is the number of house in the area dxdy at (X=x,Y=y), in this case constant

integrating gives the cdf, which you gave earlier

$$F_{X,Y}(X\leq x,Y \leq y) = \int_0^y \int_0^x f_{X,Y}(X=x,Y=y)dxdy =\int_0^y \int_0^x 2000dxdy = 2000xy$$

so instead of working in probabilty, its just shifted to number of houses