Timeline for Definition of conditional expectation for singleton
Current License: CC BY-SA 4.0
8 events
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| Jan 10, 2021 at 0:51 | comment | added | Mateusz Kwaśnicki | The symbol $E[X|Y]$ has a well-established different meaning: it is a random variable. Existence of the function $f$ is a consequence of the Doob–Dynkin lemma. It has a Wikipedia page with some references, see also this MO question for further discussion. (Apparently this is a homework problem at Cornell.) | |
| Jan 9, 2021 at 18:57 | comment | added | timudk | Also, could you point me to a proof about the existence of such a function if $Y$ is real-valued? | |
| Jan 9, 2021 at 14:42 | comment | added | timudk | Oh that helps a lot; one question though: why would one not write $E[X | Y](y)$ just like we write $\sin(x)$? | |
| Jan 8, 2021 at 20:26 | history | edited | Carlo Beenakker | CC BY-SA 4.0 |
typo
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| Jan 8, 2021 at 20:12 | comment | added | Mateusz Kwaśnicki | What I want to say is that $E[X|Y=y]$ defines a function of $y$ (or, more precisely, it is an expression that defines a function of the free variable $y$, up to equality almost everywhere with respect to the distribution of $Y$). For a fixed $y$, this is typically not well-defined. Note that in a similar way, one commonly writes e.g. $\sin x$ for a function of $x$ rather than the value for a particular $x$. Just a useful short-hand notation. | |
| Jan 8, 2021 at 17:57 | comment | added | timudk | Are you saying that the value $y \in \mathbb{R}$ is irrelevant? | |
| Jan 8, 2021 at 14:37 | comment | added | Mateusz Kwaśnicki | This usually denotes a measurable function $f$ such that $E[X|Y] = f(Y)$. (Under the usual conditions — in particular if $Y$ is real-valued — such a function always exists, and it is defined up to a set of zero measure with respect to the distribution of $Y$.) | |
| Jan 8, 2021 at 14:07 | history | asked | timudk | CC BY-SA 4.0 |