Convolution Integral Explained - Understand Fundamentals

[barksdalemc]

Can someone explain convolution to me. I have read three different books and gone to office hours and am not getting the fundamentals.

 

[quasar987]

In what context? Do you mean you don't understand some of the "applications"?

 

[barksdalemc]

I'm trying to understand in the context of probability distributions. What the convolution of the sum of two random variables represents.

 

[quasar987]

Oh.

I never really took time to ponder about this. The way it was presented to me was that the convolution appeared kind of coincidentally:

We set out to find the density f of Z=X+Y by finding it's repartition function F and then differentiating it. So we proceed from definition

$$F_{Z}(z)=P(X+Y<z)=\int_{-\infty}^{+\infty}\int_{-\infty}^{z-y}f_X(x)f)Y(y)dxdy=\int_{-\infty}^{+\infty}F_X(z-y)f_Y(y)dy$$

This is the convolution $F_X$ and $f_Y$. The density of Z is found simply by differentiating $F_Z$ wrt z and it gives the convolution of $f_X$ and $f_Y$.

There is probably a way to understand something from this and gain some insights about the relation btw the sum of two random variables.

Let me know if you find something interesting.