# Can we always write a random variable as conditional expectation plus error?

Consider the random variables $$Y,X$$. I believe that we can always write $$Y=E(Y|X)+\epsilon$$ with $$E(\epsilon|X)=0$$.

Question: Is the above true regardless whether $$Y$$ is a discrete or continuous random variable?

My thoughts: I believe that the answer is yes.

For example, let $$Z$$ be a discrete random variable and $$Y\equiv \mathbb{1}\{Z=3\}$$, where $$\mathbb{1}\{\cdots\}$$ is the indicator function taking value $$1$$ if the condition inside is satisfied and zero otherwise. We can write $$\mathbb{1}\{Z=3\}= E(\mathbb{1}\{Z=3\}| X)+ \epsilon \quad E(\epsilon|X)=0$$ that is $$\mathbb{1}\{Z=3\}= Pr(Z=3|X)+ \epsilon \quad E(\epsilon|X)=0$$

• Sure you can--but this decomposition is not always meaningful or mathematically helpful. For instance, let $X$ have a Gamma distribution and $Y$ have a Poisson$(X)$ distribution. The possible values $\epsilon$ can take on (all with positive probability) are the possible values of $Y-E(Y\mid X) = 0-X, 1-X, 2-X, \ldots.$ When $X$ is not integral (and it has zero chance of being an integer) this isn't a particularly nice set of values. Even worse, these sets differ for all $X,$ meaning there's not a lot of commonality among the $\epsilon.$
– whuber
Feb 11, 2021 at 18:59
• Wait, $E[A|B]=0$ only if but not if $E[A]=0$ right?
– BCLC
Jan 17, 2022 at 9:47

Yes this always possible, as long as the expectations exist:

$$E(Y|X) = E(Y|X).$$ $$E(Y|X) + Y= E(Y|X) + Y.$$ $$Y= E(Y|X) + Y - E(Y|X).$$ $$Y= E(Y|X) + \epsilon.$$

We then have $$E(\epsilon) = E(Y - E(Y|X)) = E(Y) - E(E(Y|X)) = E(Y) - E(Y) = 0$$ by linearity of expectations and the law of iterated expectations.

Note that in general and especially when $$Y$$ is discrete, $$E(Y|X)$$ will be non-linear in $$X$$.

• Thanks. Is $E(\epsilon|X)=0$ by construction?
– TEX
Feb 11, 2021 at 16:59
• Well, as I show it follows from linearity of expectations and the LIE, which follow from probability axioms and further definitions, so yes, in this sense (and I believe this is the "common sense"), it holds by constructing/defining $E(\epsilon|X)$ as $Y - E(Y|X)$. Feb 11, 2021 at 17:02
• 1 - ah so this holds even if $Y$ is neither continuous nor discrete? 2 - Wait, $E[A|B]=0$ only if but not if $E[A]=0$ right?
– BCLC
Jan 17, 2022 at 9:44
• AFAIK there are no random variables that are something else than continuous or discrete. Your second statement is not true. Note that in my answer, by definition, $\epsilon = Y - E(Y|X)$, so that $E(\epsilon|X) = E(Y|X) - E(Y|X) = 0$. Jan 18, 2022 at 10:36