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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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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31 views

Decomposition of sum of two independent random variables [on hold]

Let $x$ and $y$ be two independent random variables. How can I get an expression for: $$E[x|x+y=a]$$ where $a$ is a constant? In other words, is there a general rule to recover the expected value of ...
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22 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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11 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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22 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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47 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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17 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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24 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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100 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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200 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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60 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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45 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 ...
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52 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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23 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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89 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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257 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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64 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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258 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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83 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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36 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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78 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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19 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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48 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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48 views

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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65 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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21 views

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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70 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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166 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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180 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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87 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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98 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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50 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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84 views

Slope of a Regression

Given this chart, how do you determine the slope of the regression Y on F?
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88 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. $ ...
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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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251 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 ...
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266 views

Law of total variance as Pythagorean theorem

Assume $X$ and $Y$ have finite second moment. In the Hilbert space of random variables with second finite moment (with inner product of $T_1,T_2$ defined by $E(T_1T_2)$, $||T||^2=E(T^2)$), we may ...
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185 views

Proof for E[X|X] = X

I saw from the lecture that $$E[X|X]=X$$ where $X$ is a random variable. But when I am trying to prove it formally, I cannot reach to this conclusion. $$E[X|X]=\int{xf_{x|x}(x) ...
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35 views

Expression for conditional expectation

Suppose we have two random variables: $X$ is a continuous r.v.; $Y$ is a discrete r.v. taking values $0$ and $1$. Is the following expression true? $E[(E[X|Y])^{2}]= [(E[X|Y=1])^{2}]\times P(Y=1)+ ...
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191 views

Confusion related to expectation notation

I have a confusion regarding notation of the expectation. Consider the following: $$E_x(x^2) = \sum_x x^2p(x)$$ $$E_x(x|y) = \sum_x xp(x|y)$$ So the same notation here is giving different meaning ...
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Mathematical definition of causality

Let $Y$ and $X$ be random variables. $E(Y|X)$ is the conditional mean of $Y$ given $X$. We say $Y$ is not causally related to $X$ if $E(Y|X)$ does not depend on $X$, which implies it is equal to ...
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116 views

Conditional vs unconditional expectation

I'm having trouble understanding the calculation of conditional versus unconditional expectations in this case: \begin{array}{c c c c} & \quad &X=1\quad &X=-1\quad &X=2 \\ ...
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61 views

can Dirichlet prior distribution be larger than 1?

This question is related to my quest of clustering the sequences using mixture Markov modeling. I have trouble understanding Dirichlet priors in the context of MAP-estimate (Mixture Markov Models). ...
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49 views

Is the following property for positive random variables fulfilled in general?

[I have cross-posted this from math.stackexchange: http://math.stackexchange.com/questions/476466/is-the-following-property-for-positive-random-variables-fulfilled-in-general ] Suppose we have a ...
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77 views

Conditional Expectaction 3 variables

Suppose $X,Y$ and $Z$ are multivariate normal with means and full covariance matrix. The conditional expectation $E(X | Y)$ is well know. What is the conditional expectation of $E(X | Y,Z)$ if $Y$ and ...
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E-step in EM-algorithm using MAP estimate (mixed Markov models), what does it calculate?

I am trying to grasp what exactly is "estimated" in the E-step of the algorithm. According to all definitions, in E-step the "conditional expectation values , or posterior probabilities of the ...
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450 views

Expected number I will be on after drawing cards until I get an ace, 2, 3, and so forth

I am having some trouble solving the following. You draw cards from a standard 52-card deck without replacement until you get an ace. You draw from what is remaining until you get a 2. You continue ...
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22 views

Martingale difference vs. sub-independence

We know that martingale difference is an assumption weaker than dependence, and stronger than uncorrelated. The same is true for sub-independence. My question is how sub-independence and martingale ...
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64 views

Expectation of truncated distribution with two random variables in conditional

How to find the conditional expectation $E[A_1|A_1\ge A_m,A_2 \ge A_m,A_1+A_2 \ge 2A_y]$ where $0 \le A_1 \le 1$; $0 \le A_2 \le 1$; $\frac{1}{2} < A_m < A_y < 1$; $A_m, A_y$ are constants; ...