# Convolution: Addition of independent random variables

When are independent variables and we can easily calculate the mean and variance of as

(1)

Many times however we are also interested in other characteristics of . This information is incorporated in its distribution function and definitely in its density function . The question addressed here is to derive whenever the density functions of , respectively, and are known. Independence does NOT imply that . Note that, we then have and rescaling by a won’t save us either. To derive the density for explicitly we use the definition of density functions, ie, that function for which we have

(2)

As depends on both the values of we have to condition on either of these variables, in this case, . Let us use these ideas as follows:

(3)

where the equality on the second line is due to independence of and as usual we have written for the distribution function of . The density function for is thus given by differentiation with respect to .

(4)

where we have used the symmetry to derive the latter. Hence, if we know the form of we can mix these density with each other, regardless of order, to derive the density function .