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Y(y) = P(Y y) = P(g(X) y) = Z A(y) p X(x)dx where A(y) = fx g(x) yg The density is p Y(y) = F0 Y (y) If gis strictly monotonic, then p Y(y) = p X(h(y)) dh(y) dy where h= g 1 Example 3 Let p X(x) = e x for x>0 Hence F X(x) = 1 e x Let Y = g(X) = logX Then F Y(y) = P(Y y) = P(log(X) y) = P(X e y) = F X(e) = 1 e e y and p Y(y) = eye e y forThat is, the events {X ∈ A} and {Y ∈ B} are independent events (b) Let g(x) be a function only of x and h(y) be a function only of y Then E(g(X)h(Y)) = (Eg(X))(Eh(Y)) 4# & ' !
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