Warmup Problem for Problem Set 6 (18.440)
Let X ∼ Unif[0, 1) and Y ∼ Unif[0, 2) be independent uniform random variables. Find the pdf of Z = X +Y .
What shape does the pdf function have? Why?
Hint 1: Draw the picture to find out the bounds of the integral to get the cdf FZ . Then find the pdf fZ .
Also, note that FZ (t) and fZ (t) are only piecewise smooth, so you will need to compute them separately on
each of the intervals over which they are smooth.
Hint 2: Alternatively, you can use convolution. Remember, fZ (t) is still only piecewise smooth, so you
will need to take this into account when computing the convolution.
1