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Consider the random variable $X$ on $(\Omega,\mathcal{F},P)$ and let $\mathcal{G}\subset\mathcal{F}$.

Q1. What is the difference between $E[X]=E[X|\{\emptyset,\Omega\}]$ and $E[X|\sigma(X)]$?

Q2. Which of $E[X|\mathcal{F}]$ and $E[X|\mathcal{G}]$ is "more random"?

Thanks...

Freakalien
  • 229

1 Answers1

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Q1. $E[X|\{\emptyset,\Omega\}]=E[X]; \ E[X|\sigma(X)]=XE[1|\sigma(X)]=X$

Q2. $\mathcal{G}\subset\mathcal{F}$. So, $\mathcal{G}$ is coarser "information" and the first conditional expectation is "more random" in this sense. For example, take $\mathcal{G}=\{\emptyset,\Omega\}$.

  • Just to be absolutely clear you mean $E[X|\mathcal{F}]$ is in some sense more random than $E[X|\mathcal{G}]$? – Freakalien Apr 21 '14 at 04:43
  • Yes, conditional expectation is a random variable. So, conditioning on $\mathcal{F}$ yields $X$ and conditioning on trivial sigma field ${\emptyset,\Omega}$ yields a constant which is not random –  Apr 21 '14 at 04:49