Thia post is a guided example of a practice problem (Problem 15A found in Exam P Practice Problem 15 – Still More Convolution Practice). The exposition is to make clear the thought process on how to set up and evaluate the integral to find the pdf of an independent sum. The following is the statement of the problem:
Find the pdf of where and are independent random variables such that is uniformly distributed on the interval and is exponentially distributed with mean 10.
Setting up the Scene
The pdfs of and are:
Since and are independent, the joint pdf of and is . The following figure shows the support of this joint distribution.
Figure 1 – Support of the Joint Distribution
Let denote the pdf of the independent sum . To calculate , we need to sum the joint density over the entire green region in Figure 1. Each value of is the sum of the joint density on a particular line (see Figure 2 below).
For each point of the line , the joint density at that point is or . Thus we have:
Let’s focus on integral . Since the support of the variable is a bounded interval, there are two cases to consider: and . So we calculate separately in each case. Let's look at the following figures.
Looking at Two Cases
Figure 3 – Case 1
Figure 4 – Case 2
Case 1 covers all the lines from the red line to the blue line in Figure 3. Case 2 covers all lines above the blue line in Figure 4. These considerations will dictate the limits for the integral in .
Setting Up Integrals
Here’s the integrals for the two cases.
The following two figures show how integrals and are set up. For a typical line in Case 1 (Figure 5), the range for is . For a typical line in Case 2 (Figure 6), the range for is .
Figure 5 – Typical Line in Case 1
Figure 6 – Typical Line in Case 2
Thus we have the following pdf:
More information can be found in Examples of convolution (continuous case).
Additional practice problems can be found in: