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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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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21 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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9 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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37 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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15 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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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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42 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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12 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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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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72 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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26 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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101 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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73 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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28 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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40 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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21 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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22 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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28 views

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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21 views

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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74 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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49 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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56 views

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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48 views

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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68 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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39 views

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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50 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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62 views

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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102 views

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, ...
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28 views

Correlation of particle patterns, mutual density and basic statistics

I am a postgrad student, who managed not to see any statistics in his Maths and Physics undergrad degree (am I the only one surprised by this ?). Anyway, I wanted to ask if someone could explain the ...
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Conditional Distribution of an Independent Variable for missing data

Let $X=[X_1 X_2 X_3 ... X_p] $be a matrix of p independent variables where $X_i=[x_{i1} ... x_{in}]'$ is a nx1 vector. Let W be a nxn weight matrix based upon queen contiguity (so zero's along the ...
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51 views

Calculating the mean excess loss

Suppose $X$ has the following pdf: $$ f_x(x)=0.01 \qquad for\space 0\le x<100$$ Find the pdf of $X_p$ (the excess-loss variable) and calculate the mean excess loss for $d=10$. \begin{align} ...
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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!
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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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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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279 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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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: ...
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127 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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197 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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26 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 ...
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64 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 ...