Question: 1.Problem10.27LetXbeacontinuousrandomvariablewithprobabilityden sityfunction f(x).Supposethat theprobabilitydistributionfunctionF(x)= P(Xx)isstrictlyincreasingontherangeofX.DefinethefunctionI(u)as theinversefunctionofF(x).Verifythat (a) P(I(U)x)=P(Xx)forallx,wherethecontinuousrandom variableUisuniformlydistributedon(0,1). (b)Foranyfunctiong(x),E[g(X)]= g(x)f(x)dx= 1 0 h(u)du,where thefunctionh(u)isdefinedbyh(u)=g(I(u))for0

1.Problem10.27LetXbeacontinuousrandomvariablewithprobabilityden sityfunction f(x).Supposethat theprobabilitydistributionfunctionF(x)=

P(X≤x)isstrictlyincreasingontherangeofX.DefinethefunctionI(u)as theinversefunctionofF(x).Verifythat

(a) P(I(U)≤x)=P(X≤x)forallx,wherethecontinuousrandom variableUisuniformlydistributedon(0,1).

(b)Foranyfunctiong(x),E[g(X)]= ∞

−∞g(x)f(x)dx= 1 0 h(u)du,where thefunctionh(u)isdefinedbyh(u)=g(I(u))for0

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