Hi everyone,
I'm working on a project where I need to find a way to directly convolve two cumulative distribution functions (CDFs) given in polynomial form, and solve the following convolution integral:
\[ (F * G)(x) = \int F(x - y) \, dG(y) \]
Specifically, the functions I'm working with are:
\[ F(x) = 0.17 + \frac{0.83x}{150} \quad \text{for } 0 \leq x \leq 150 \]
\[ G(y) = 0.25 + \frac{0.75y}{45} \quad \text{for } 0 \leq y \leq 45 \]
For x > 150 , F(x) = 1 , and for y > 45 , G(y) = 1 , as these are cumulative distribution functions.
The real context of the problem is as follows: these CDFs describe the waiting time of a pedestrian at a traffic signal with the probability (cumulative) that they will wait, at most, that time. Therefore, the composition of both CDFs must necessarily include two points:
- (x=0, y=0.0425): This point represents the situation where the pedestrian encounters both signals open, resulting in a cumulative probability of 0.17 x 0.25 = 0.0425 .
- (x=195, y=1): This point represents the situation where the pedestrian encounters both signals closed at the moment they have just closed, resulting in a total waiting time of 150 + 45 = 195 seconds, with a cumulative probability of 1.
Does anyone know any algorithm that can accomplish this convolution, giving the answer in polynomial form? Any help or pointers would be greatly appreciated!
I've tried looking into two posts - this one and this one - but neither offers a systematic method. One provides a formula with a function inside the differential dG(y) , and I'm having difficulty understanding how this can be expanded, hence my search for a code alternative.
Thanks in advance for your help!