Cumulative Distribution Function of Distance

[Kalinka35]

Homework Statement

The location of an emergency is uniformly distributed over a city district. The district is a square rotated 45 degrees with "radius" r (the distance from the center to the top corner is r).
When the emergency occurs, the ambulance is at the center of the district. Let D be the "right-angle" distance from the ambulance to the emergency.
I need to find the cumulative distribution function F_D(d) and the probability density function f_D(d).

Homework Equations

The Attempt at a Solution

I know that the PDF is the derivative of the CDF so I am trying to find the CDF first.
It seems like you could define two random variables X and Y that give the coordinates of the emergency and the sum of those would give you the right-angle distance, but that seems like it might be overcomplicating things and you might just be able to work with D, the total distance.
Either way I just don't see how you can obtain a CDF.

 

[lanedance]

so you could look at it this way...

the area of any region, divided by the total area of the square is the probability of emergency happening in that area

so consider a infinitesimal circular strip, between s & s + ds. You could work out the area of the strip to get the pdf of the distance

the only trick being when the circle exceeds the square at given regions. Then just work out the angle of the circle that is actually within the square

Though you could also try the combination of random varibles as you suggest, I'm not sure how it would work out until I tried, could be clean...