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Suppose X and Y are independent and X ~ gammala, y) and Y ~ gamma(6, 7)

Prove the following

a) U=X+Y~gamma(α + β,γ)

b) V=X/(X + Y ) ∼ beta(α,β)

c) U, V independent

d) W^{2}~gamma(1/2, 1/2) when W~N(0,1)

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Answer #1

Since Xgamna (Y) and Y~amma (,v) we have Since x and Y ave inde pendenty dishibuled ai fevential i en by compeund pehab.litySince W No,1 ie. if x NCa NOJ with paf N The distributien function G f W= ib given by Go)PWEu) = P (w 2w) PEAwWE here No P(WE

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