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March 23rd, 2010, 09:49 AM   #1
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Joint probability distribution of random variables

Let X and Y be two independent exponential random variables with distinct parameters ? and ?. Find the density of X+Y. I know that for the single variable case the exponential random variable is usually given by f(x)=?e^(-?x) if x>0 and f(x)=0 for x<0. I'm unsure how to find the density (i know I need to integrate both for X and Y) but I don't know how to combine these to find the density X+Y. Any help is greatly appreciated. Thanks!
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March 23rd, 2010, 05:47 PM   #2
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Re: Joint probability distribution of random variables

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Originally Posted by meph1st0pheles
Let X and Y be two independent exponential random variables with distinct parameters ? and ?. Find the density of X+Y. I know that for the single variable case the exponential random variable is usually given by f(x)=?e^(-?x) if x>0 and f(x)=0 for x<0. I'm unsure how to find the density (i know I need to integrate both for X and Y) but I don't know how to combine these to find the density X+Y. Any help is greatly appreciated. Thanks!
The general formula to get the density function for the sum of two independent random variables is the convolution.

Let f(x) and g(x) be the density functions of X and Y. Let h(x) be the density function of X+Y.
Then: h(x)= ?g(u)f(x-u)du.
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