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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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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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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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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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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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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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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70 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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47 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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108 views

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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2answers
39 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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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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62 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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36 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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45 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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79 views

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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52 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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43 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, ...
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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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40 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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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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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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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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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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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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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] - ...
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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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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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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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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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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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32 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 ...
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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 ...
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160 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
115 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 ...
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73 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 ...