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More precisely let $(X,Y)$ is a pair of continuous random variables with joint density function $f(x,y)$ and we assume $\mathbb{E}(|Y|) < +\infty$. Define $$H(X) = \int\limits_{-\infty}^\infty \frac{f(x,y)}{f_X(x)}y dy,$$ where $f_X(x) = \int\limits_{-\infty}^\infty f(x,y) dy$.

For $H(X)$ to satisfy the definition we have to show that

  • $\mathbb{E}(|H(X)|) < +\infty$,
  • $H(X)$ is $\sigma(X)$-measurable and
  • $\mathbb{E}(Y \cdot I_A) = \mathbb{E}(H(X) \cdot I_A)$, for every $A \in \sigma(X)$, where $I_A$ is the indicator of $A$.

This is a homework assignment for me, so I only need some hints, not a full solution.

Faragó Dávid
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    The second property you want to show is rather obvious. $x \mapsto H(x)$ is continuous. So you have composition of two measurable mappings: $\omega \mapsto X(\omega) \mapsto H(X(\omega))$ – Calculon Sep 12 '16 at 08:09
  • Not sure if this would be the right way but for the last property you could perhaps use a $\pi-\lambda$ argument. – Calculon Sep 12 '16 at 08:18

1 Answers1

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Hint on third bullet:

On base of $H\left(x\right)f_{X}\left(x\right)=\int f\left(x,y\right)ydy$ we find for suitable functions $g$:

$$\mathbb{E}\left[H\left(X\right)g\left(X\right)\right]=\int H\left(x\right)g\left(x\right)f_{X}\left(x\right)dx=\int\int f\left(x,y\right)yg\left(x\right)dydx=\mathbb{E}\left[Yg\left(X\right)\right]$$

drhab
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  • This really helped, thanks. Could you also tell me what makes a function $g$ suitable? – Faragó Dávid Sep 12 '16 at 09:02
  • It must be a Borelmeasurable function such that the expectation of $H(X)g(X)$ exists. If the expectation of $H(X)$ exists then it is enough if $g$ is measurable and bounded, so then it can be applied on functions like $1_A$. – drhab Sep 12 '16 at 09:57
  • You are very welcome. – drhab Sep 12 '16 at 10:02