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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:

**Problem 15A**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.

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**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).

**Figure 2**

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.

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**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 .

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**Setting Up Integrals**

Here’s the integrals for the two cases.

**Case 1 **

**Case 2 **

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**

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**Calculation**

**Case 1**

**Case 2**

Thus we have the following pdf:

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**Additional Practice**

More information can be found in Examples of convolution (continuous case).

Additional practice problems can be found in:

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I can’t tell you how fantastic it is. With less than a week to go until my exam I embarrassingly spent several HOURS on this kind of problem. I stumbled upon this only after having come up with it on my own, but it is a good confirmation. Readers, PAY ATTENTION! This is good info!!!