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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What is the relationship between orthogonality 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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Mean of predictive distribution

I observe independent, Poisson-distributed data $ D = \{x_1, ... x_n \} $ with mean parameter $ \mu $. Over $ \mu $ I assume $ Gamma(\alpha_0, \beta_0) $ as a prior (where $ \alpha_0 $ and $ \beta_0 $ ...
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28 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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33 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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74 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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42 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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38 views

$Y = \beta_0+\beta_1*X+U$ 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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41 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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90 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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2answers
87 views

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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16 views

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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31 views

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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23 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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12 views

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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25 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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58 views

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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54 views

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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247 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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36 views

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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81 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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61 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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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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27 views

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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72 views

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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84 views

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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29 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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131 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
95 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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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 ...
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70 views

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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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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59 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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107 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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1answer
15 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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91 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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24 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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117 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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130 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 ...
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249 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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38 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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11 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 ...
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1answer
84 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 ...