Consider generating a random variable Y by the following mechanism.

Nov 12th, 2015
KateS
Category:
Engineering
Price: $10 USD

Question description

Consider generating a random variable Y by the following mechanism. First, generate X = x distributed as a Gamma(α, λ) random variable, then generate Y as an exponential with rate parameter x. 

  • (a)  Find P(Y<1).
  • (b)  Find the conditional distribution of X given Y = y.
  • (c)  Find the conditional expectation and variance of X given Y = y (if you carefully examine your answer for part (b), you won’t need to do any integrals).
  • (d)  Are X and Y independent random variables? Give both an intuitive and mathematical justification for your reasoning. 


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