Probability density function

[aaaa202]

If g and f are two normalized probability density functions is it then true in general that the convolution of f and g is normalized too?

 

[mathman]

Yes. The convolution is the density function for the sum of the random variables which have g and f as density functions.

 

[aaaa202]

how do you show that normalization is preserved..

 

[mathman]

how do you show that normalization is preserved..

Let f(x) = g(x)*h(x) (where * means convolution). ∫f(x)dx = ∫g(x)dx∫h(y-x)dy. Let u = y-x for the h integral (remember these integrals are over the entire real line) and you have the product of 2 integrals, each of which is 1.