Question

Let ? and ? be two independent uniform random variables such that ?∼????(0,1) and ?∼????(0,1).

A) Using the convolution formula, find the pdf ??(?) of the random variable ?=?+?, and graph it.

B) What is the moment generating function of ??

Answer 1

convolution formulae for Z=X+Y is given by: ( where X and Y are 2 continuous random variable)

Range of Z will be from 0 to 2

Now let us apply this general formula to our particular case. We will have for z<0 and also for z≥2. Now we deal with the interval from 0 to 2. It is useful to break this down into two cases (i) and (ii) 1<z<2

(i) The product is 1 in some places, and 0 elsewhere. We want to make sure we avoid calling it 1 when it is 0. In order to have, we need z−x≥0, that is, x<=z. So for (i), we will be integrating from x=0 to x=z. we have

(ii) Suppose that 1<z<2. In order to have to be 1, we need z−x<=1 that is, we need x>=z-1 So for (ii) we integrate from z−1 to 1. We have,

Graph :

B)