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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Double expected value, which comes first?

In the following equation, the outer expectation is over the distribution $X_i|T_i = 1$ $\tau|_{T = 1} = E(E(Y_i|X_i, T_i = 1) - E(Y_i|X_i, T_i = 0)|T_i =1)$ Are we taking the expected value of ...
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30 views

Conditional Expectation of a product and sum of three Gaussian Random Variables

I have a problem and I have no idea how to tackle it. Any help would be greatly appreciated. The problem is the following: Assume I have three mutually independent random variables $a$, $b$, and $c$, ...
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49 views

E[x|y] where y=cdf(x) and y is random

I have a grid approximation of a cdf, $F_x$. The cdf has support for $x>=0$ From there, calculating the $E[x]$ is straight forward with some std numerical integration techniques. In my case, ...
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100 views

Conditional expectation everywhere non-zero while unconditional one zero?

I have a real-valued random variable $X$ that takes on positive and negative values, and $$E(X)=0 \tag{A}$$ There is also another real-valued random variable $Y$, not independent from $X$, neither ...
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106 views

Order Statistics problem: why doesn't law of total expectation (Adam's law) work?

This is the problem The opening prices per share, $Y_1$ and $Y_2$, of two similar stocks are independent random variables, each with a density function given by $$f (y) = ...
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115 views

How can I maximize my chances on an exam?

I have an exam tomorrow, there are 18 topics, the professor gives only one. But it is not enitrely random how he gives the topic. First he gives an interval like 18-52, 30-62, 30-68 etc, then we ...
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47 views

How to compute $P(|X - E_Y[h(y)]| < c)$?

Consider a discrete random variable $Y$, a continuous random variable $X$, and a constant $c$. The goal is to find $$P(|X - E_Y[h(y)]| < c),$$ when we are only given $P(y)$, function $h(y)$, and ...
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24 views

Conditional moments of bivariate normal

Suppose that (X,Y) are bivariate normal with non-zero means and correlation. Is there any neat expression for $\mathbb{E}(X|Y>0)$?
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113 views

Is it true that $E[Y|W]=E[Y|E(X|W)]$, given that $W$ is $X$ measured with error

In Carroll's 2006 book "Measurement Error in Nonlinear Models", and on p38 it is stated without proof that: [One can] estimate the regression of $Y$ on $(Z, X)$ and then to substitute into this ...
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281 views

Conditional Expectation E[X] = E[X|Y<=a] + E[X|Y>a]

Is it generally true that for random variables $X$ and $Y$, regardless of being dependent or independent, that $E[X] = E[X \mid Y \le a] + E[X \mid Y>a]$ ?
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131 views

Regression Specification and Relation to Conditional Expectation Function

In their book, Mostly Harmless Econometrics, Angrist & Pischke introduce regression as an approximation to the conditional expectation function (CEF). They present (p 46) this equation: ...
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Minimizing absolute devitation using median

I need to prove that the expectation of absolute deviation is minimized by the median. We are given that $$med(y|x) = \beta_0 + \beta_1x$$ and $x$ can take on three values with positive probability: ...
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45 views

Conditional Expectation of Order Statistics

Given $X_1,...,X_n \sim f(x)$ How do I find $E(X_{(1)} | X_{(2)})$? Would I have to find the conditional pdf and integrate wrt x? I get the conditional distribution to be $f_{X|Y}(x|y) ...
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23 views

Finding the best predictor Brownian motion

I want to find the best predictor of $(B_3-B_2)(B_4-B_{\pi})$ given an observation of $B_1$ Where $B_t$ is brownian motion for time $t \geq 0$. I am not sure how to approach this. I know it will be ...
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11 views

choice of maximum likelihood over expectation maximisation

Given a probability distribution two common statistical measures are the expectation value and the maximum likelihood (equivalent to mean and mode?). My question is, given a probability distribution ...
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40 views

Expected number of trials

Consider independent trials, each of which is a success with probability p and derive the expected number of trials needed to obtain k consecutive successes by (a)conditioning on the time of the ...
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24 views

A step of a proof regarding the Nadaraya-Watson estimator

Let the data be $(y_i , X_i) $ where $y_i$ is real valued and $X_i$ is a q-vector. The regression function for $y_i$ on $X_i$ is $g(x) = E(y_i | X_i = x)$, we can write this as: $$y_i = g(X_i) + ...
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16 views

Conditional expectation in mixture distributions

I have a mixture distribution for observed lifetime data $(\delta_i,t_i,L_i)$, where $\delta_i$ is a censoring variable (1 indicating death, and 0 indicating censoring), $t_i$ is the observed lifetime ...
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92 views

Nadaraya-Watson Optimal Bandwidth

I am currently working on a statistical project where I need to estimate a conditional expectation $E[Y|X=x_i]$ using the Nadaraya-Watson estimator. For doing that, I have the sample ...
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14 views

Conditional Expectation for Forecasting Intervention Model

William and Wei: Time Series Analysis Univariate and Multivariate Methods, Second Edition, page 90, gives the conditional expectation for $Z_{n+l}$ ($l$ step forecast of $Z_n$): $$ \hat{Z}_n(l) = ...
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17 views

Mean distance of the point

Two points $X_1$ and $X_2$ are uniformly distributed in $D_{2r}(O)$, the disk of radius 2r centered at the origin. What is the expectation of $|X_2|$ given that one of the vertices (WLOG $X_1$) lies ...
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96 views

Comparison of two estimators based on mean squared error

I'm refreshing my knowledge of statistics, and I'm stuck on a problem that asks for a comparison between 2 estimators (unfortunately, I can't remember the source of the problem, but I do know its ...
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29 views

Conditional pmf and mean for geometric rv

I have a geometric r.v. p(1-p)^(n-1), n = 1, 2, 3, 4, ... I was able to figure out the conditional mean conditioned on X > a. My answer to this is E[X] + a. I am pretty sure that is correct but if ...
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144 views

An 'easy' exercise on conditional expectations and filtrations

I am struggling with the following exercise in the context of modeling information structure via filtration to evaluate contingent claims. I hope that someone can explain me how to derive the ...
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106 views

Marginal, joint, and conditional distributions of a multivariate normal

Let $Y$ ~ $MVN_3(\mu, \Sigma)$ where $\mu = (5,6,7)$ and $\Sigma = \begin{bmatrix}2 & 0 & 1\\0 & 3 & 2\\1&2&4\end{bmatrix}$ Find (a) The marginal distribution of $Y_1$ (b) The ...
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34 views

Cumulative distribution function of dependent variables

I have measured packet delays on several different radio channels $c_1, c_2, c_3$ and got three streams of delay data: $d_{11}, d_{12}, d_{13}, \dots$ $d_{21}, d_{22}, d_{23}, \dots$ $d_{31}, ...
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45 views

Simple question about finding conditional expectation

Let $X,Y \sim U[0,1]$ ($X,Y$ are independent), we want to find $E[X|X>Y].$ I tried a few approaches to the above problem, but am not confident in my answer. One approach is as follows. Note that ...
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23 views

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 ...
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34 views

Conditional Expectation and Variance with Multiple Conditions

Question on the properties of conditional expectation - Is it true that $E [W(t) | W(s) - W(u)] = E[W(t) | W(s), W(u)]$ ? Context - To prove that for a Wiener process $W(t)$, $E[W(t) | W(s),W(u)] = ...
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Mulitvariate normal truncated conditional expectation

There is three-variable multivariate normal distribution. Denote 3 variables with $X_1$, $X_2$, $X_3$. Let $\mu_i$ be means, and $\sigma_i^2$ variances of respective variables, and let $\Sigma$ be ...
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Estimating conditional variance y|x

I am building a predictor for $y = f(x)$ using training samples ${(x_i, y_i)}$ (assume) drawn i.i.d from some distribution $p(x,y)$, by optimising the empirical L2-loss: $f(x) = argmin_f \; \sum_i ...
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How to compute the expectation of a normally distributed random variable given an imprecise signal?

Given $r\sim\mathcal{N}\left(\bar{r},\frac{1}{\alpha}\right)$ where $0<\bar{r}<1$ and an imprecise signal about $r$, $x_i=r+\epsilon_i$ where ...
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75 views

Conditioning on Two Variables

Does the following equality hold true? $E[Z|\{X,Z\}] = Z$ If not, then when will it hold true?
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Estimate E[x|A,B]: alternatives to bucketing for non-parametric estimation

I have a set of products. I would like to estimate Expected Value of items sold of the products wrt product price and age of the purchaser. One alternative is to assume a distribution and fit it. ...
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50 views

Can E[X|Y] be computed from E[X|Y,Z]?

Let X, Y and Z be 3 discrete random variables. Is the following true? $\sum_{i=1}^\infty P\{Y=i\} E[X|Y=i,Z] = E[X|Z]$
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Expectation on higher-order products of normal distributions

I have two normally distributed variables $X_1$ and $X_2$ with mean zero and covariance matrix $\Sigma$. I am interested in trying to calculate the value of $E[X_1^2 X_2^2]$ in terms of the entries of ...
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What is the relationship between orthogonality and the expectation of the product of RVs

Is there such thing as a statistical concept of orthogonality? Does somebody could provide a formal explanation about the relationship between orthogonality and conditional expectation of a RV? Here ...
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Quadratic model as linear decrease in proportions

Assume $Y_i <= X_i$ for all $i$. The conditional expectation of our data was found to satisfy $E[Y|X=x] = a1*x-a2*x^2$ to very good accuracy for a large range, with $0<a2<a1<1$. ...
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87 views

Mean of predictive distribution

I observe independent, Poisson-distributed data $ D = \{x_1, ... x_n \} $ with mean parameter $ \mu $, i.e., $$x_i\stackrel{\text{iid}}{\sim}\mathcal{P}(\mu)$$ Over $ \mu $ I assume $ Gamma(\alpha_0, ...
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Quadratic mean convergence of a biased coin using conditional expectation

I'm a master's degree student and after a lot of research and some days trying I still can't get the answer for a question proposed by my Statistics professor. He asks to toss a coin with a random ...
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54 views

Prove conditional expectation $E[X|X>x]$ is the unconditional expectation $E_{P^*}[X]$ under a probability measure $P^*$

Prove that the conditional expectation $\mathbb E[X|X>x]$ (here x is fixed, say x=10) is the unconditional expectation $\mathbb E_{\mathbb P^*}[X]$ under a probability measure $\mathbb P^*$. Derive ...
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Conditional expectation of $\mathbb{E}(X - Y | (X, Y)\in\mathcal{A})$

Given two independent random variables $X \sim \mathcal{U}[-1,5]$ and $Y \sim \mathcal{U}[-5,5]$, what is $$\mathbb{E}\{Y - X | X \le 1, Y > X, Y \in [-1,1] \}\,?$$ I managed to compute the ...
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Law of iterated expectations with two random variables

Let $X$ and $Y$ be two random variables. I want to calculate $E[X|X<Y]$. I am wondering whether I can use the law of iterated expectations in order to calculate it, i.e. $E[E[X|X<Y,Y]]$. Do I ...
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$Y = \beta_0+\beta_1*X+U$ and $W = \gamma_0+\gamma_1*X+\gamma_2*U$, assume $\gamma_2\neq0$. also given is $E(U|X) = E(U)$ . find $ E(U|W,X)$

$Y = \beta_0+\beta_1*X+U$ and $W = \gamma_0+\gamma_1*X+\gamma_2*U$, assume $\gamma_2\neq0$. also given is $E(U|X) = E(U)$ find $ E(U|W,X)$ and conditions under which $E(U|W=w,X=x)$ is an increasing ...
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How to estimate the correlated individual components from a sum, for a random process?

Assume that there are $N$ realisations of five individual, random variables$X_1$, $X_2$, $X_3$, $X_4$ and $X_5$, which in general could be correlated. We define another random variable ...
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45 views

Question about an expectation

Let $x$ and $\gamma$ be vectors. Here it says that $$E[y-x'\gamma]^2 = E[(y-E[y|x])^2 + (E[y|x]-x'\gamma)^2]$$ However, I don't see why $$E[(y-E[y|x])(E[y|x]-x'\gamma)] = 0.$$ By the way, $E$ is the ...
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Random variables with some properties (conditional expectation)

I am looking for two random variables which fulfills the following two things: a) $\mathbb E(X|Y)<\infty$ and $\mathbb E(Y|X)<\infty$ b) $E(X|Y)> Y$ and $\mathbb E(Y|X)>X$ a.s Here is ...
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A random variable that induces a $\sigma$-algebra the same as the one in the sample space

Consider a probability space $(\Omega, \mathcal{F}, P)$ where $\Omega$ is the sample space, $\mathcal{F}$ is the $\sigma$-algebra of $\Omega$, and $P$ is the probability measure. Let ...
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Expectation maximisation for right-censored iid data from Normal

This is the data (which are length of ropes), $\textrm{Data}=\{99, 70, o ,89, 88, o, 88,70, o ,o\}$, where $o$ are censored data with value above $100$. Assume that data are from $\textrm{iid} \sim ...
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A question regarding symmetry properties of a uniform distribution [duplicate]

Was anyone able to explain why $$E(U_2) = 0$$ I don't quite understand what the relevance of the underlined statement - "by the symmetry of $U_1$" in determining $E(U_2)$ is edit: I get it now, ...