grandes-ecoles 2025 Q17

grandes-ecoles · France · mines-ponts-maths1__mp Continuous Probability Distributions and Random Variables Verification of Probability Measure or Inner Product Properties

Let $\left( X _ { i } \right) _ { i \in \mathbf { N } }$ be a sequence of independent random variables all following a Rademacher distribution. Show that the map $\varphi$ defined on $\left( L ^ { 0 } ( \Omega ) \right) ^ { 2 }$ by $$\forall X , Y \in L ^ { 0 } ( \Omega ) , \quad \varphi ( X , Y ) = \mathbf { E } ( X Y )$$ is an inner product on $L ^ { 0 } ( \Omega )$.

Let $\left( X _ { i } \right) _ { i \in \mathbf { N } }$ be a sequence of independent random variables all following a Rademacher distribution. Show that the map $\varphi$ defined on $\left( L ^ { 0 } ( \Omega ) \right) ^ { 2 }$ by
$$\forall X , Y \in L ^ { 0 } ( \Omega ) , \quad \varphi ( X , Y ) = \mathbf { E } ( X Y )$$
is an inner product on $L ^ { 0 } ( \Omega )$.

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