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Chapter 8: Q. 8.13 (page 393)

Show that ifE[X]<0androle="math" localid="1649871241073" 胃鈮�0is such thatEe胃X=1, then胃>0.

Short Answer

Expert verified

Consider the functionf(y)=-ln(y),y>0. Since this function is convex, if we letY=e胃X, using Jensen's inequalityE[f(Y)]鈮�f(E[Y]),

Step by step solution

01

Given Information.

E[X]<0and 胃鈮�0is such thatEe胃X=1, then胃>0.

02

Explanation.

Assume thatE[X]<0and 胃鈮�0such thatEe胃X=1, whereby 胃is a scalar. We want to show that胃>0. To do this, we will use Jensen's inequality. It says that for a convex function fand an arbitrary random variable Yis

E[f(Y)]鈮�f(E[Y])

Consider the functionf(y)=-ln(y),y>0. Since this function is convex, if we letY=e胃X, using Jensen's inequality, we get:

E-lne胃X鈮�-lnEe胃X

Sincelneu=u, for eachu, and using the informationEe胃X=1, we get:

E[-胃X]鈮�-ln(1)=0

if properties of expectation

-胃E[X]鈮�0

鈬�it is givenE[X]<0

-胃<0

-胃<0

胃>0.

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