MHB Calculating the Density Function for X/Y with Exponential Distributions

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The discussion focuses on calculating the density function for the ratio of two independent random variables, X and Y, both following exponential distributions. The users suggest using transformations, specifically u = X + Y and v = X/Y, to facilitate the calculation. It is noted that X has a distribution of λe^(-λx) and Y has a distribution of θe^(-θy). A participant provides insight that Y/X has a distribution of (θ/λ)e^(λx - θy). The conversation emphasizes the importance of transformations in deriving the desired density function.
yinon
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X,Y r.v statistically independent ,with exponential Distribution.
calculate the density function of X/Y

(Let $X$ have distribution ${\lambda}e^{-{\lambda}x}$ and $Y$ have distribution ${\lambda}e^{-{\lambda}y}$

i know i should use transformtion u=X+Y ;v=X/Y to solve it)
 
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Hello and welcome to MHB, yinon! :D

Can you show us what you have tried so our helpers know where you are stuck and how best to help?
 
Let $X$ have distribution ${\lambda}e^{-{\lambda}x}$ and $Y$ have distribution ${\theta}e^{-{\theta}y}$. Then $Y/X$ has distribution $\frac{\theta}{\lambda}e^{{\lambda}x-{\theta}y}$
 
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