In Theorem 2.1.10 the probability integral transform was proved, relating the uniform cdf to any continuous cdf.
Question:
(a) Prove that Y is stochastically greater than a uniform(0,1); that is, if U ~ uniform (0,1), then
P(Y > y) ≥ P(U > y) = 1 -y, for all y, 0 < y < 1,
P(Y > y) ≥ P(U > y) = 1 - y, for some y, 0 < y < 1.
(Recall that stochastically greater was defined in Exercise 1.49.)
(b) Equivalently, show that the cdf of Y satisfies Fy(y) ≤ y for all 0 < y < 1 and FY(y) < y for some 0 < y < 1.
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a We prove part b which is equivalent to part a b Let Ay x F x x y Since F x ...View the full answer
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