Let X and Y be independent random variables, X ~ Gamma(α,λ) and Y ~ Gamma(β,λ). Prove tha S=X+Y andT =X/(X+Y) are independent ,S~Gamma(α+β,λ) and T ∼Beta(α,β).
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1What have you tried? Where are you stuck? – JimB Oct 15 '20 at 02:42
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I used moment generating function to prove that X + Y has a Gamma(a+b,lambda). I stuck at how to prove T~Beta(a,b) and their independency. – lookyourphone Oct 15 '20 at 02:57
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1Use the Jacobian formula for the joint distribution of $(S,T)$. – Xi'an Oct 15 '20 at 03:47