Calculating Density Function of Joint Independent Exponential Random Variables

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SUMMARY

The discussion focuses on calculating the density function of the ratio Z = X / Y, where X and Y are independent exponential random variables with potentially different parameters. Participants explore the process of deriving the probability density function (pdf) of Z by first expressing it in terms of Y and then averaging over all possible values of Y. Key equations and methods for integration are discussed, emphasizing the importance of understanding the properties of exponential distributions in this context.

PREREQUISITES
  • Understanding of exponential random variables and their properties
  • Knowledge of probability density functions (pdf)
  • Familiarity with integration techniques in probability theory
  • Basic concepts of joint distributions and independence in statistics
NEXT STEPS
  • Study the derivation of the pdf for the ratio of two independent random variables
  • Learn about the convolution of probability distributions
  • Explore the use of moment-generating functions in analyzing random variables
  • Investigate applications of exponential distributions in real-world scenarios
USEFUL FOR

Statisticians, data scientists, and anyone involved in probability theory or statistical modeling, particularly those interested in the behavior of ratios of random variables.

MATHTAM
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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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