MathOverflow

The problem
Assume $p > 1$. Consider the function$$f(x,y) = x^p y^{1-p}, \qquad x,y > 0.$$Note that$$f'' = p(p-1)x^{p-2}y^{-1-p}\begin{bmatrix}y \\ & x\end{bmatrix}\begin{bmatrix}1 & -1 \\-1 & 1\end{bmatrix}\begin{bmatrix}y \\ & x\end{bmatrix...

Proof -
Clearly for an equally likely case, if there are n elements {a,b,c...} P(a)=1/n and P(a) can be zero only if n is infinite.
For non equally likely cases, suppose the sample space is S={a,b} and (P(a)/j)=(P(b)/k), where j and k are integers. Then, as the events are not equally likely then we can change the frequency with which they occur, let the new sample space be like,...

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