If

is the PDF of a random variable

, then

,

, (the convolution of

and

). This means that finding the PDF of

involves computing the

-fold convolution of

with itself, a computationally intensive operation to do directly even for small

and simple PDF functions. This can be made easier in some cases by using the convolution property of Fourier transforms,

, and the observation that the characteristic function of

is the same as the Fourier transform

, which means that

can be computed by rescaling the inverse Fourier Transform of

. For the uniform distribution, this lets us compute the distribution for

much more quickly than we could directly, though it is still a little slow for larger

.