# Exam P Practice Problem 14 – More Convolution Practice

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Problem 14A
Suppose $X$ and $Y$ are independent random variables such that $X$ is exponentially distributed with mean 10 and $Y$ has the following density function:

$\displaystyle g(y)=e^{-0.2 \ y}-e^{-0.25 \ y}$ where $y>0$

Find the pdf of $W=X+Y$.

Problem 14B
Suppose $X$ and $Y$ are independent random variables such that $X$ is exponentially distributed with mean 20 and $Y$ has the following density function:

$\displaystyle g(y)=\frac{1}{3} \ e^{-0.1 \ y}-e^{-0.2 \ y}+\frac{2}{3} \ e^{-0.25 \ y}$ where $y>0$

Find the pdf of $W=X+Y$.

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The two problems here provide more practice for using convolution method to find the pdf of an independent sum. An actual Exam P question may not be this straightforward. But using convolution to find the pdf of a sum is a good skill to have. The problems here can help solidify the thought process behind the method of convolution.

For a discussion of the convolution technique, see the blog post Examples of convolution (continuous case).

For additional practice, see Exam P Practice Problem 13 – Convolution Technique.

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