Linked Questions

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1answer
6k views

iterated expectation conditional on two variables

How to prove that $E[Y]=E[E[E[Y|X_1, X_2]]]$ ? PS. I don't see how $E[E(Y|X_{1},X_{2})|X_{1}]=Y[Y|X_{1}]$ and $E[Y]=E[E(Y|X_{1})]$ can be used here. But it feels close. Please help, I'm stuck PPS. ...
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1answer
1k views

Exact meaning of conditional expectation $\mathbb{E}[X|\mathcal{F}]$

I'm going through elementary literature on measure theory from Shreve (Vol II) and having a hard interpreting the meaning of $\mathbb{E}[X|\mathcal{F(t)}]$ where $X$ is a random variable and $\mathcal{...
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1answer
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Law of iterated expectations for several variables

There are tons of questions related to the LIE, but all the ones I've seen do not help in my case, including this one and this one. I know that by Law of Iterated Expectations (LIE), $E(x_{i}|A_{i})=...
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1answer
861 views

Covariance of Poisson and Conditional Binomial RV's

Problem Statement Let $X$ and $Y$ be random variables such that $X \sim \text{Poisson}(\lambda)$ and $Y|X \sim \text{Binomial}(x+1,p)$. Find $\text{Cov(X,Y)}$. Attempt at a Solution I would like to ...
4
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1answer
794 views

Are the law of iterated expectation and the law of total expectations the same?

On the Wikipedia page of the Law of total expectations it is said that The proposition in probability theory known as the law of total expectation, the law of iterated expectations, the tower rule, ...
2
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2answers
447 views

Is this formula for the Law of Iterated Expectations correct?

I saw two versions of the law of iterated expectations, this one: \begin{align} E(E(Y\vert X)) = E(Y) \end{align} and this one: \begin{align} E(E(Y\vert X_1, X_2)\vert X_1) = E(Y \vert X_1) \end{align}...
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1answer
476 views

Implications of strict exogeneity for OLS in time series

Zero Conditional Mean (ZCM), or Strict Exogeneity, is given by: $E[u|X]=0$ Equivalently, $E[u_t|X]=0, t=1,...,T$ Is it true that this implies: Zero Unconditional Mean: $E[u_t]=0, \forall t$ ...
2
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1answer
255 views

Proving for an AR(2) process that $E[X_t | F_{t-1}]=E[X_t | F_{t-2}]=E[X_t | F_{t-3}]$

An exercise states: Using the law of iterated expectations applied to an AR(2) process, verify that $E_{t−k} . . . E_{t−1} (X_t ) = E(X_t |F_{t−k} ) $ for $ k = 1, 2, 3 $ where $ E_{t−k} (X_t ) = E(...
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1answer
89 views

Conditional Expectation with two random variables

I know $E[E[x\mid y]] = E[x]$ by smoothing property of the expectations. Then, I came across the following equation: In this equation we have: $$E[E[x\mid A,Z]\mid Z] = E[x\mid Z]$$ I try to ...
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1answer
90 views

Does $E(X \mid Y,Z)=0$ imply $E(X \mid Y)=0$?

Does $E(X \mid Y,Z)=0$ imply $E(X \mid Y)=0$? In other words, if we have $E(X \mid Y,Z)=0$ then can we also say $E(X \mid Y)=0$?