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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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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+50

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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25 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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72 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 ...
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25 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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17 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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115 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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4answers
95 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 ...
1
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1answer
38 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
29 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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10 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 ...
6
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1answer
81 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 ...
6
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322 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
39 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
28 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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13 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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53 views

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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21 views

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
30 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
101 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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219 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
97 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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62 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
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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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33 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, ...
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99 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 $N$ ...
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270 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 ...
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68 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 ...
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310 views

Expected value of sample median given the sample mean

Let $Y$ denote the median and let $\bar{X}$ denote the mean, of a random sample of size $n=2k+1$ from a distribution that is $N(\mu,\sigma^2)$. How can I compute $E(Y|\bar{X}=\bar{x})$? Intuitively, ...
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105 views

Conditional Expectation of Poisson Random Variables

Suppose $X_1,X_2,\ldots,X_n$ is a random sample from a Poisson Distribution with mean $\theta$. How can I find the conditional expectation $E \left( X_1+X_2+3X_3 |\sum_{i=1}^n X_i \right)$? I know ...
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52 views

Conditional heteroskedasticity/variance and uncertainty in estimated residuals

Say you've got a simple cross-sectional model $$ y=\alpha + \beta T +\epsilon $$ where $T$ is binary. This is basically a t-test, but lets treat it like a regression. You're interested in the ...
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114 views

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 ...
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37 views

Martingale and deterministic functions

Suppose: $u_t \sim N(0,1) \ iid.$, $X_t = g(X_{t-1}) \cdot u_t$ whereas $g(X)$ can be any deterministic function. Is this sufficient to define a martingale? So does it hold: $E(X_t|X_{t-1}, \ldots , ...
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How to model conditional moments using an additive model

Let's say that I have the data set $(X,Y)$ where $X$ is a p-dimensional variable and $Y$ is uni dimensional. I'm interested in the following model: $$ \theta_y = E(Y|X) \\ \theta_{y^2} = ...
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53 views

What is the distinction between covariance and condtional sampling variance for two random variables in case of meta-analysis of sample correlations?

It seems (I have a vague notion) that there is some similarity between the two measures. Alternatively stated, there is certain relationship between the two? could you please clarify the idea
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conditional expectation of squared standard normal

Let $A,B$ independent standard normals. What is $E(A^2|A+B)$? Is the following ok? $A,B$ iid and hence $(A^2,A+B),(B^2,A+B)$ iid. Therefore we have $\int_M A^2 dP = \int_M B^2 dP$ for every ...
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77 views

Conditional expectations by conditioning on functions of random variables

I have conjectured the following: Let $f:\mathbb{R}\supseteq A \rightarrow B \subseteq \mathbb{R}$ be an injective function. Let $X$ be a random variable with support $A$ and $Y$ be some random ...
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Showing that E(E(Y|A))|B) = E(Y|A,B)?

Would $E( E(Y|A) | B) = E(Y|A,B)$? How does one show this? Intuitively, it make sense that conditioning on this one at a time would result in the same thing as conditioning all at once.
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81 views

Is expectation of univariate variable dependent on the multivariate

Suppose I have a mutivariate Gaussian distributed variable $u\sim\mathcal{N}(\mu,\Sigma)$, where $\Sigma$ is a dense matrix. I wish to calculate the expectation of $f(u_i)$. Is $$E(f(u_i))=\int ...
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239 views

Conditional expectation subscript notation

This should be a relatively simple question. I'm trying to confirm my understanding of the subscript notation on expectations when the subscript denotes a conditioning. In the example ...
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213 views

Variance of sample mean of bootstrap sample

Let $X_{1},...,X_{n}$be distinct observations (no ties). Let $X_{1}^{*},...,X_{n}^{*}$denote a bootstrap sample (a sample from the empirical CDF) and let ...
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1answer
114 views

Covariance with conditional expectation

Suppose $X$ and $Y$ are random variables, $E(Y^2) < \infty$ and $\varepsilon = Y - E(Y|X)$ so that $Y = E(Y|X) + \varepsilon$. Given that $E(\varepsilon | X) = E(\varepsilon) = 0$, show ...
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112 views

How to compute conditional expectations with respect to a sigma field?

Example: Toss a coin twice. Letting $\mathbb P$ be a probability measure, suppose $\mathbb P(HH)=p^2,\mathbb P(HT)=\mathbb P(TH)=p(1-p), \mathbb P(TT)=(1-p)^2.$ I would like to answer the following ...
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53 views

Show that the best mean square estimator of $X$ given $(X_{1},…,X_{n})$ is $\hat X =E[X|\sigma(X_{1},…,X_{n})]$

Let $X$ and $X_{i}$, $i=1,...,n$ be random variables on a probability space $(\Omega , \mathcal F,P)$. Show that the best mean square estimator of $X$ given $(X_{1},...,X_{n})$ is $\hat X ...
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88 views

Slope of a Regression

Given this chart, how do you determine the slope of the regression Y on F?
5
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1answer
91 views

If $X_{n+1}$ is a martingale subject to $Y_0,\ldots,Y_n$, then is it a martingale with respect to $Y_0^2,\ldots,Y_n^2$?

I don't have a very solid foundation in measure theory, and this always seems a bit confusing to me so I would appreciate any help. We are given $ E \left( X_{n+1} | Y_0,\ldots,Y_n \right) = X_n. $ ...
13
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1answer
827 views

Subscript notation in expectations

What is the exact meaning of the subscript notation $\mathbb{E}_X[f(X)]$ in conditional expectations in the framework of measure theory ? These subscripts do not appear in the definition of ...
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328 views

Problem with proof of Conditional expectation as best predictor

I have an issue with the proof of $E(Y|X) \in \arg \min_{g(X)} E\Big[\big(Y - g(X)\big)^2\Big]$ which very likely reveal a deeper misunderstanding of expectations and conditional ...