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I don't know if I can find someone here to help me understand this issue, but I'll try

the jensen inequality can be found here http://en.wikipedia.org/wiki/Jensen%27s_inequality

I have the following discrete random variable [itex]X[/itex] with the following pmf:

x 0 1 2 3

pr(x) 0.110521 0.359341 0.389447 0.140691

and the summation is (1).

defining the function [itex]Y=\frac{1}{X}[/itex]

Then finding the [itex]\bar{X}=\sum_{i=0}^{3}{iPr(i)=1.560308}[/itex]

Now calculating [itex]E[Y]=E[\frac{1}{X}]=\sum_{i=1}^{3}{\frac{1}{i}Pr(i)}=0.6009615[/itex]

then:

[itex]Y(\bar{X})=\frac{1}{\bar{X}}=\frac{1}{1.560308}=0.640899105[/itex]

According to Jensen inequality what should happen:

[itex]E[\frac{1}{X}] \geq \frac{1}{E[X]}[/itex]

but what I got is the opposite case!!!!!!!!

Now also I have to mention the function [itex]Y=\frac{1}{X}[/itex] is convex given that the second derivative is strictly larger than zero.

Do I misunderstand the Jensen inequality? or is there something wrong with the convixity assumption?

Any idea?

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# Jensen inequality, unexplained distribution, very confusing problem

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