Calculating Density Function of Joint Independent Exponential Random Variables

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The discussion focuses on finding the probability density function (pdf) of the ratio Z = X / Y, where X and Y are independent exponential random variables with different parameters. Participants suggest starting by determining the conditional pdf of Z given Y=y, which involves using the transformation of variables technique. The next step involves averaging this conditional pdf over the distribution of Y to obtain the marginal pdf of Z. Key equations and integration techniques are discussed to facilitate this calculation. The overall goal is to derive a clear expression for the density function of the ratio of two independent exponential random variables.
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X and Y are independent, exponentially distributed random variables - with possibly different parameters

Determine the density func. of Z = X / Y

How to attack ?
 
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What eqn can you write for the pdf of Z at z given Y=y?
How can you then average that out over all possible y?
 

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