Convolution of Two Densities

t

density function f

Gaussian

uniform

tent

parabolic

density function g

Gaussian

uniform

tent

parabolic

The convolution of two functions can be thought of as a measure of the overlap of the graphs as one graph is shifted horizontally across the other. Formally, if

f

and

g

are functions, the convolution of the two is the function

(f*g)(t)=

∞

∫

-∞

f(t-τ)g(τ)dτ

.

The plot shows

f(t-τ)

, that is,

f

shifted by

t

units, in blue,

g(τ)

in purple, and the product of the two in gold. Thus the gray area is exactly the value of the convolution at

t

.

If

X

and

Y

are independent random variables with respective density functions

f

and

g

, then the density function of

X+Y

is the convolution of

f

and

g

. Interestingly, the convolution of two Gaussian densities is a Gaussian density.