grandes-ecoles 2017 Q8

grandes-ecoles · France · x-ens-maths2__mp Continuous Probability Distributions and Random Variables Entropy, Information, or Log-Sobolev Functional Analysis
Show that for all $f \in \mathscr { C } _ { b } ^ { 2 }$, $f$ admits an entropy relative to $\mu$ and that $$\operatorname { Ent } _ { \mu } ( f ) \leqslant \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } \mu ( x ) d x$$ You may consider the family of functions defined by $f _ { \delta } = \delta + f ^ { 2 }$ for $\delta > 0$. 
Show that for all $f \in \mathscr { C } _ { b } ^ { 2 }$, $f$ admits an entropy relative to $\mu$ and that
$$\operatorname { Ent } _ { \mu } ( f ) \leqslant \int \left| f ^ { \prime } ( x ) \right| ^ { 2 } \mu ( x ) d x$$
You may consider the family of functions defined by $f _ { \delta } = \delta + f ^ { 2 }$ for $\delta > 0$.
Paper Questions
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