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

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Can someone provide an proof for $E[P[A|X]] = P[A]$

I'm tired of seeing the word "trivial" for this equality on every single lecture notes I could find online. Can someone please show me why this is indeed trivial? Thank you!
2
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1answer
45 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 ) = ...
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25 views

Proof of alternating conditional expectation base equations

How do we prove the base equations for Alternating Conditional Expectation algorithm. The statement is thus: We define arbitrary mean-zero transformation $\theta(Y),\phi(X_i)$,$1<i<p$ for ...
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4answers
220 views

What is the difference between $E(X|Y)$ and $E(X|Y=y)$?

Generally, What is difference between $E(X|Y)$ and $E(X|Y=y)$? Former is function of $y$ and latter is function of $x$? It's so confusing..
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28 views

Most probable value given observation

Suppose I have observed $Z = 3$, where $Z = X + Y$, where $X \sim N(0,9), Y \sim N(0,4)$. How would I find the most probable value of $X$ that would have given me $Z = 3$? My attempt at a solution: ...
3
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1answer
66 views

Find conditional expectation given a discrete random variable whose range is N

Consider the following random variables in $(\Omega, \mathfrak{F}, P)$. a $X_1,X_2, X_3,...$ where $\forall n \in \mathbb{N}, \mu_n = E(X_n), \sigma_n^2 = Var(X_n) < \infty$ b $N$, a discrete RV ...
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42 views

Conditional vs. Unconditional Maximum Likelihood

I have some questions on the difference between conditional MLE (CMLE) and unconditional MLE (UMLE) in practice. In what follows I will only talk about the unconditional and conditional mean and leave ...
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25 views

Is it always true that $E[E[X|Y]^2] = E[X|Y]^2$? [duplicate]

X and Y are random variables. So $E[X|Y]$ is conditioned on a random variable. Do we always have: $$E[E[X|Y]^2] = E[X|Y]^2.$$ I have the doubt because I know that $E[X|Y]$ is a random variable ...
2
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0answers
53 views

On $E[E[Y|X]|X]= E[Y|X]$

I am trying to simplify $E[YE[Y|X]|X]$ can I use this property: $$E[E[Y|X]|X]= E[Y|X]$$ If yes I have never seen a Proof of this property (that seems very reasonable), could I have a reference? If ...
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28 views

on the minimization of: $E[((Y-f(X))^2|X]$ [duplicate]

I am having troubles solving this exercise: Deduce that the random variable $f(X)$ that minimizes $E[((Y-f(X))^2|X]$ is $$f(X)= E[X|Y]. $$ I proceeded in this way: $$E[(Y-f(X) + E[Y|X] - ...
2
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1answer
23 views

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?
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votes
2answers
69 views

conditional expectations value

I need to calculate the following integral $$\int_{\mu+c}^{\infty} y\cdot \frac{1}{\sigma\sqrt{2\pi}}e^{(y-\mu-w)^2/2\sigma^2}dy$$ So essentially $y\sim N (\mu+w, \sigma^2)$ and im trying to ...
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78 views

conditional expectations

Hi i was wondering how to figure out the following Suppose $y=x+e$ where e is an i.i.d error. Say $x \sim N(\mu,\sigma_1^2)$ and $e \sim N (0, \sigma_e^2)$ which means $y \sim N (\mu, ...
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77 views

How do I solve $E\left[ E \left(X|Z \right) E\left( Y|Z \right)\right]$?

I am trying to solve $E\left[ E \left( \mathbf{X}|\mathbf{Z} \right) E \left( \mathbf{Y}|\mathbf{Z} \right) \right]$, (where $\mathbf{X}$, $\mathbf{Y}$, and $\mathbf{Z}$ are random variables) but I am ...
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80 views

Finding the value of the pressure to maximize mean

The peel strength ($Y$) of photo resist in the lamination process of PCBs depends on the pressure ($x$) and temperature ($z$) by the relation: $$Y = 50 + 5x + 10z + (20xz)-(.01(x^2)\cdot (z^2))$$ ...
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1answer
27 views

How to work with conditional expectations and variances?

I have seen this statement in a lecture note : $$E[(Y-f(X))^2\,|\,X]=Var[Y|X]+E^2[(Y-f(X)\,|\,X]]$$ If $Z,X$ are random variables, then shouldn't be $Var[Z|X]=E[Z^2|X]-E^2[Z|X]$?In which case ...
6
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3answers
173 views

Simulation involving conditioning on sum of random variables

I was reading this question, and thought about simulating the required quantity. The problem is as follows: If $A$ and $B$ are iid standard normal, what is $E(A^2|A+B)$? So I want to simulate ...
3
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2answers
108 views

Conditional expectation of $X$ given $Z = X + Y$

Suppose I have two independent normal variables $X$ and $Y$ with known mean and variance. Defining $Z = X+Y$, what is the most straightforward way to compute $\mathbb{E}\left[X|Z\right]$? I am ...
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1answer
75 views

Best statistical notation for expected probability density

Assume that we have two multivariate normal distributions $\mathcal{N}_1 = \mathcal{N}(\mu_1, \Sigma_1)$ and $\mathcal{N}_2 = \mathcal{N}(\mu_2, \Sigma_2)$. We do these two steps: Pick a point, say ...
3
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1answer
72 views

How to find $E[x|y]$ when distributions of y and x are separately known,(p.s. they are both Gaussian)?

In detail, I have these relations (in order of causality): $u_1 = ax_0$ $x_1 = u_1 + x_0$ $y = x_1 + w$ where $w = N(0,1), x_0 = N(0,\sigma^2)$. This was my approach: I know the distribution of ...
2
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1answer
59 views

Bounded expectation implied bounded conditional or vice versa?

If $\mathrm{E}\left(X\right)<\infty$ does that imply $\mathrm{E}\left(X|Y\right)<\infty$? How about vice versa? I'm thinking if we condition on an event (say $Y>2$) then if we have ...
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61 views

Predicted probabilities from simulated betas and hypothetical data after conditional logit?

I'm working with conditional a conditional logit model to avoid bias that comes with FE logit models, when it comes to generating some hypothetical substantive effects, however, I run into trouble. ...
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43 views

Law of iterated expectations - an small exercise

From edX MIT probability course - Widgets and Crates: Let $X_i$ be the number of widgets in a particular box $i$. Let $N$ be the number of boxes in a crate. Assume $X$ and $N$ are independent, ...
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99 views

Conditional Expectation via Integral over Quantile Function

Following this thread "Does a univariate random variable's mean always equal the integral of its quantile function?" I tried to do a similar thing for a conditional expectation. It seems like my ...
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8 views

Understanding the conditioning in a GARCH process

In a GARCH model like the following $$y_t=\sigma_tz_t,\\ \sigma_t^2=\omega(1-\alpha-\beta)+\alpha y_{t-1}^2+\beta \sigma_{t-1}^2$$ where $z_t$ is assumed to be iidN(0,1), we say that conditional on ...
4
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1answer
43 views

Expectation of conditional normal distribution

I have two jointly normally distributed variables $s_1$ and $s_2$. I am now searching for the conditional expectation $$ E(s_1|s_1>r_1,\ s_2>r_2) $$ where $r_1$ and $r_2$ are constants. An idea ...
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19 views

Give an example about to conditional expectations with respect to a sigma field

I have problem with exercise, I didn't solve. Give an example on $\Omega=\{a,b,c\}$ in which $$E(E(X|F_1)|F_2)\neq E(E(X|F_2)|F_1)$$ Thanks very much for your help.
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116 views

How to find this integral [duplicate]

Let $X_1, \cdots, X_n$ be $iid$ normal random variables with unknown mean $\mu$ and known variance $\sigma^2$. How to find $E[\Phi(\bar X)]$, where $\bar X:=\frac{\sum_{i=1}^nX_i}{n}$, please? I guess ...
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115 views

Expected value of q given y is weighted average of mean q and and y

It is assumed that: 1) $y=q+u$ Where $q$ is productivity and $y$ a testscore that measures true productivity. $u$ is a normally distributed error term, independent of $q$, with zero mean and ...
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1answer
87 views

Covariance of conditionally independent random variables

I want to find the covariance between two random variables $X$ and $Y$, which are independent given another random variable $M$. I thought the calculation would be $$cov[X,Y]=E_M[cov[X,Y|M]] ...
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1answer
35 views

Conditional expectation of random vector given low-rank linear transform

Given the random vector $$ \mathbf{h} = \left(\begin{matrix} \mu \\ \varepsilon_1 \\ \varepsilon_2 \end{matrix} \right) \thicksim \mathcal{N}(\mathbf{0}, \mathbf{\Sigma}_\mathbf{h}) $$ and the ...
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11 views

Expectation of RV conditional on being larger than others

probably dumb question, but I am not quite sure here. I want to calculate $\mathbb{E}[X_i|X_i > X_j \ \forall j \neq i]$ note, that this is not the expectation of the maximum distribution as I ...
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1answer
84 views

Can someone give a clear-cut idea of $E(X|X<Y)$?

If $X$ and $Y$ be two i.i.d. random variables, then what should $E(X|X<Y)$ essentially look like? P.S.-Is the denominator equal to $\frac12$? (the two r.v.'s being iid serving as the motivation ...
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665 views

A generalization of the Law of Iterated Expectations

I recently came across this identity: $$E \left[ E \left(y|x,z \right) |x \right] =E \left(y | x \right)$$ I am of course familiar with the simpler version of that rule, namely that $E \left[ E ...
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1answer
43 views

Expectation of Truncated & Random Variable

I have what appears to be a relatively simple question, but am struggling to understand how to go about answering it. The general question is as follows: What is the expected value of $S_{I}$, ...
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1answer
38 views

Conditional expectation with conditioning on two independent variables

Let $Y_1$ and $Y_2$ be independent r.v.s, and $X$ be another random variable. What is $E[X|Y_1, Y_2]$? Is $E[X|Y_1, Y_2]$ equivalent to $E[X|Y_1] \cdot E[X|Y_2]$? More specifically, is there a ...
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15 views

Knowing the level of aggregate processes, how to get the levels of constituents?

I have a bunch of component processes $y_{it}$, where $i=1..n$. I can build reasonable time series models $y_{it}=f_i(y_{i,s<t},X_t)$, where $X_t$ - exogenous variables. These could be ARIMAX ...
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24 views

Probability of classification given two observations

If two doctors each correctly diagnose a disease 55% of the time, and both agree in a certain case there is disease, what is the probability of disease? Surely it is higher than 55%?
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In EM derivation why can I sum over the iid variables in the conditional expectation?

In EM when you take the expectation: $E[\log P(y,x \mid \theta)\mid x, \theta']$ $= \sum\limits_yP(y\mid x, \theta') \log P(y,x\mid \theta)$ I understand this but the following part I don't ...
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weighted conditional expectation in adaboost

I'm looking at this paper, http://www.stanford.edu/~hastie/Papers/AdditiveLogisticRegression/alr.pdf around pages 10-11 (marked 346,347 in the pdf). This notation is introduced $$ E_w[g(x,y)|x] := ...
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1answer
33 views

Recursive definition of conditional normal distribution

I am acquiring samples that are Gaussian distributed and I need to calculate $$ p(x_n | x_{n-1}, x_{n-2}, \ldots , x_1) $$ for each sample $x_n$ as it comes in. I am trying to break this expression ...
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1answer
104 views

Conditional expectation of $(X+Y)^2 | X = x$

If the density $f(x,y) = c$ when $x>0$, $y>0$, $x+y < 1$ and $0$ otherwise, find $$E((X+Y)^2 | X = x)\text{ for } x \in (0,1).$$ How to approach this question? Can we approach it by ...
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2answers
227 views

Conditional probabilities/expectations

A coin minting machine randomly produces unbalanced coins so that the probability of getting a head in tossing a coin is a random variably $Y$. Supposed $Y$ has a pdf $f(y) = 2y$ for $0 <= y <= ...
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1answer
136 views

Conditional expectation in AR(1) process

Suppose we have a stationary AR(1) process: $Y_{t+1}=a+ \rho Y_{t} + \epsilon_{t+1}$ where $\epsilon_{t+1}$ is white noise with probability density function $\phi(.)$. Now say we have a equation ...
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73 views

Validating statistical tests for value at risk and expected shortfall

I am trying to figure out if value-at-risk (VaR, a quantile) type tests could capture if expected shortfall (expectations above a quantile) point forecast generated from a type of model could be ...
3
votes
1answer
53 views

Average causal effect of one year increase in schooling vs a four-year increase in schooling

I'm not sure why in Mostly Harmless Econometrics, last paragraph of p. 55, the expectations of $f_{i}(s-4)$ is taken and the expectation of $f_{i}(s-1)$ is not. The text reads: Conditional on ...
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37 views

Can conditional Gaussian estimation problem solved by linear regression?

It just occurs to my mind about under what situation can the linear regression produce good result (valid). for example, suppose y, $x_1$, $x_2$, ... $x_n$ follow multivariate Gaussian distribution, ...
5
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1answer
117 views

Closed form recurrence formula for getting N consecutive heads on a coin

I want to find the expected number of coin tosses to get $N$ heads in a row, where $p$ is the probability of getting a head in a single toss. Let $F(N)$ be the expected number of tosses to get $N$ ...
6
votes
1answer
279 views

Conditioning on independent random variables

I am in a situation where I have to compute: $$E(u(x_1)|\bar{X},S^2)$$ where $X_1$ is a normally distributed random variable and $u(.)$ some function. I know that by the student's theorem the sample ...
4
votes
1answer
72 views

Is this statement about the conditional expectation of a sum true?

For expectations of random variables (RVs) $X$ and $Y$ it is true that $$E(X+Y)=E(X)+E(Y)$$. My question is whether when conditioning on RV vector $Z_{1...J}$, it is also true that ...