Statistical Analysis Stack Exchange
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Questions tagged [conditional-expectation]

A conditional expectation is the expectation of a random variable, given information on another variable or variables (mostly, by specifying their value).

A conditional expectation, or conditional mean, is the expectation of a random variable, given information on another variable or variables (mostly, by specifying their value). [The expected height of an adult US male is different from the expected height of an adult US male, born in 1942 to Micronesian parents.

For discrete variables, $\text{E}(X|Y=y) = \sum x \ \text{P}(X=x|Y=y)$, while for continuous variables, $\text{E}(X|Y=y)= \int x \ f_X(x|Y=y) \ dx$, where the sum and integral are over the possible values taken by $x$.

An expectation conditioned on some subset of the possible values taken by $Y$, such as $\text{E}(X\mid a < Y < b )$, say, is also a conditional expectation.

For an in-depth treatment of conditional expectations see:
Billingsley, P. (1995) "Probability and Measure", 3rd edition, John Wiley & Sons Inc., New York

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Is $E[E(X|Y)|Z] =E[X|Y,Z]$ If so, how to prove?

Does $E[E(X|Y)|Z] =E[X|Y,Z]$? Also, what about $E[E(X|Y=y)|Z=z] =E[X|Y=y,Z=z]$ I'm confused by the relations. It seems intuitively to be the case. If it is correct, how do I mathematically prove it. I've searched on this site and…
KUZ
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Conditional Expectation Constant

If the conditional expectation E(Z|X) is a constant k, what can be inferred about Z? Since this means that whatever the value of x is given, Z is always k, does this imply that E(Z) is equal to k?
statBeginner
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Law of total expectation and how prove that two variables are independent

I am confused in understanding the formal definition for independence and how does this differ from the law of total expectations. I thought that $E[u|x]=E[u]$ formally means independence of $u$ and $x$. However, when I have $E[u|x]=a$ (const) and…
Majte
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How is $E[c(X)|X] = c(X)$ derived?

I understand that this is a property of conditional expectation, but I am not too clear on how it is derived.
laoganma
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$E(X|Y)=E(X) \iff E[X^nh(Y)]=E[X^n]E[h(Y)]$

Theorem: $E(X|Y)=E(X) \iff E[Xh(Y)]=E[X]E[h(Y)]$ for all bounded measurable $h$ Does this theorem also hold for all odd moments of $X$, i.e. $E(X|Y)=E(X) \iff E[X^nh(Y)]=E[X^n]E[h(Y)]$ for all bounded measurable $h$, and $n$ is an odd integer?
Cagdas Ozgenc
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Conditional expectation to define causal effect

I'm reading these notes which are discussing the NRCM approach to analyzing causal relationships, that is to say, treats the causal inference problem like a missing data problem (where the missing data are the counter-factual events). So suppose…
Addem
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Law of iterated expectations

I think I have understood why $E[E(X_1X_2|X_1)]=E[X_1E(X_2|X_1)]$; so your support would really help me being more convinced. Do we take out the $X_1$ from the inner expected value because it is calculated with respect to the marginal distribution…
Charlie
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Does this equality hold?

Is it true that $E[XY|Z]=E[X|Z]E[Y|Z]$ if $X$ and $Y$ are independent each other, but $X$, $Y$ are not independent with $Z$? Can anyone prove this? Thank you.
Ferren
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Proving $E(g(X,Y) |Y=t) = E(g(X,t)|Y=t)$ for continuous random variables

For continuous random variables $X$ and $Y$, I want to show that $E(g(X,Y) |Y=t) = E(g(X,t)|Y=t)$. This looks simple, but I can't do it. I know that if $X$ and $Y$ were discrete then $E(g(X,Y) |Y=t) = \sum_{(x,y) \in \mathbb{Z^2}}…
Noppawee Apichonpongpan
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Is the following conditional expectation property true for four random variables?

Given random variables $W,X,Y,Z$ satisfying $E(XY)=E(E(X|Z)Y)$ and $E(XW)=E(E(X|Z)W)$, must it hold that $E(XY)=E(E(X|Z,W)Y)$? I tried the case with $X,Z,W$ being jointly gaussian, and the case with $X,Z$ being jointly gaussian with mixing parameter…
Ted Leroy
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$E[W|Z] = E[XY|Z] \stackrel{?}{=} E[X|Z]E[Y|Z] \text{ if } E[XY] = E[X]E[Y]$?

The question is in the title: $$ E[W|Z] = E[XY|Z] \stackrel{?}{=} E[X|Z]E[Y|Z] $$ $$\text{if}$$ $$ E[XY] = E[X]E[Y] $$ ??? $X$ and $Y$ are independent of each other, but neither is independent of $Z$. My basic stats are a bit rusty…
generic_user
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Is E(X|f(Y)) same as E(X|Y)?

I am trying to see whether the conditional expectaction of E(X|Y) is the same as conditional expectaction of E(X|f(Y)), where f(Y) is any function of Y which is smooth, e.g. polynomial, sin etc.
Martin
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If conditional expectation is equal to unconditional expectation does that mean the random variables are independent?

If $E\left(Y|X\right) = E(Y)$ can we state that $X$ and $Y$ are independent? And vice verse, if $X$ and $Y$ are independent can we state that $E\left(Y|X\right) = E(Y)$?
falkor86
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Equivalence sum of conditional expectations given random observations and sum of conditional expectation given order statistics

Suppose $X_1,...,X_n$ are independent and identically distributed random variables defined on some probability space $(\Omega, \mathcal{A}, P)$. Define $Y=\sum_{i=1}^{n}X_i$. If we denote the corresponding order statistics as $X_{n,n} \geq X_{n-1,n}…
Joogs
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