Questions tagged [conditional-expectation]

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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AR(1) Process and Infinite Geometric Series

I'm doing an assignment and I've been told that ln(real consumption) has a unit root and is an I(1) process and that real consumption is given by: I've also been told that: Part A I need to derive ...
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Why is E[X+Y|X,Y]=X+Y?

Intuitively, it seems obvious, but I am struggling to prove it for the case where $X_1, ..., X_n$ are continuous random variables. I am aware that $E[c(X)|X]=c(X)$. So how would one show that $E[c(X_i)...
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Conditional Expectation conditioned on two independent random variables

What can we say about E(X|Y,Z) in general, where no other information is given other than the fact that Y and Z are independent? Note that, X might be dependent on both Y and Z.
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Expected value of X given X+Y=s?

Question: Given $X\sim N(\mu_X, \sigma_X^2)$ and $Y\sim N(\mu_Y, \sigma_Y^2)$ are independent, and you know $X+Y=s$. What is the expected value of $X$? I encountered this during an interview. My ...
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When mathematical statistics outsmarts probability theory

This is not a question, but it is too good to pass. I read it is originally due to Enis, Peter. "On the relation $E (X) = E [E (X∣ Y)]$." Biometrika 60, no. 2 (1973): 432-433. Assume $Y$ has ...
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Where does the random come in for conditional expectations $\mathbb{E}[X | \mathcal{F}]$?

For continuous random variables $X, Y$ the conditional expectation $\mathbb{E}[X | Y]$ is itself a random variable. I understood this in the sense that for a realisation of $Y$ we can say $$ \mathbb{E}...
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Simulation to estimate a conditional expectation

Let X and Y be independent and identical exponential random variables with parameter $\theta>0$. Compute $P[X \leq x | X+Y]$ for $x\geq0$. I tried to solve this theoretically here (https://math....
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Simple linear regression with stochastic regressors formulation through conditional expectation

Just recently I discovered there could be deterministic and stochastic regressors. Could somebody please correct me if my following reasoning is off? The conditional expectation $\mathbb{E}[Y|X]$ ...
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Is the following conditional expectation property true for four random variables?

Given random variables $W,X,Y,Z$ satisfying $E(XY)=E(E(X|Z)Y)$ and $E(XW)=E(E(X|Z)W)$, must it hold that $E(XY)=E(E(X|Z,W)Y)$? I tried the case with $X,Z,W$ being jointly gaussian, and the case with $...
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What is the derivation for "Partial Expectation"?

On the Wikipedia page for Log-normal distribution It is written that $$E[X|X>k]Pr(X>k)=\int_{k}^{\infty}xf_{X}(x)dx$$ I know it is probably simple, but I am still wondering the derivation. Since ...
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Implications of mean independence

Let $U$ be a random variable with mean $0$. Take other two random variables $X,Y$. Assume $$ (1)\quad E(U|X,Y)=0. $$ I believe (1) implies $$ E(U\cdot X)=E(U \cdot Y)=0. $$ Does (1) imply $$ E(U \cdot ...
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sum of expected differences in time series

I have a (markov) decision process without reward. I am estimating the expected state differences $\mathbb{E}[\delta_t]$ where $\delta_t = X_{t+1} - X_t$. A state $X_{t+1}$ can be expressed as $X_0 + \...
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Conditional expectation with multiple random variables

I'm trying to understand the solution to a problem from A First Course in Probability (Ross): There are two misshapen coins in a box; their probabilities for landing on heads when they are flipped ...
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What is the expectation of the exponential distribution multiplied by indicator function?

I am reading the research paper [A New Bayesian Lasso], where $u$ has the distribution The expectation of $u_j$ is given by $$\frac{1}{\lambda}+|\beta_j|$$. I know that the term $\frac{1}{\lambda}$ ...
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Continuity of conditional expectations?

An elementary result in probability theory is the so-called continuity of probability. Specifically, let $E_1\supseteq E_2\supseteq\cdots$ be a sequence of nested events. Then $P(\cap_n E_n) = \lim_{n\...
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Is the expected value of a probability over an interval meaningful?

I am reading an unpublished manuscript and have come across an equation of the following form for the calculation of the probability of an even A, $$ P[A]=E\Big[P[X>x|Y]\Big]. \tag{1} \label{1} $$ ...
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Find the conditional expectation in the following setting

Given: $X_{i} = \Theta +W_{i}$ $\Theta \sim \mathcal{N}(\mu, \sigma_{0}^2)$ $W_{i} \sim \mathcal{N}(0, \sigma_{i}^2)$ $\Theta, W_{i}$ are all independent I need to find the least mean squared ...
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A Condition a Conditional Probability Should Satisfy

Consider four random variables $V_1,V_2,V_3, V_4$. Here, suppose that $V_2$ is a function of $V_3$ and $V_4$, say $V_2=V_3-V_4$. In this case, I want to know whether the following conditional ...
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Multivariate Probability Distribution with Linear Conditional Expectation

I want to know what probability distribution has the linearity property of the conditional expectation. To be specific, suppose that we have three random variables named $v_1,\;v_2,\;v_3$. Then, if $[...
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Quick short question about understanding of conditional expectation

let me just ask one simple question, I am not sure if I understand this concept of conditioning w.r.t. sub-$\sigma$-algebras. Let $(\Omega,\mathcal{A},\mathbb{P})$ be probability space and $X,Y:\Omega\...
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Correct conditional expectation with respect to a different measurable space

Suppose we've got random variables $X_1:(\Omega_1,\mathcal{A}_1)\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R})),X_2:(\Omega_2,\mathcal{A}_2)\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R}))$ on a ...
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Expectation of truncated distribution

Consider the random variables $X,Y$ and assume that $$ E(X|Y)=0 $$ Does this imply that $$E(X|X\geq A,Y)\neq 0 ?$$ I think this holds for the truncated Normal, for example. But does it hold ...
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Does $E(X|X\in A)=\frac{E(X\mathbf{1}(X\in A))}{Pr(X\in A)}$ hold?

Does $E(X|X\in A)=\frac{E(X\mathbf{1}(X\in A))}{Pr(X\in A)}$ hold? (Here $\mathbf{1}(\cdot)$ is the indicator function). To me it seems that it holds. Here is the proof: $E(X|X\in A)=\int_{-\infty}^{\...
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What is the meaning of a quantile regression model that predicts the conditional mean?

What does that phrase "quantile regression model that predicts the conditional mean" mean? How to interpret that? I found it in Liu et al. (2020). The authors have compared the results of ...
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calculate combination of matrix of probabilities (win rate ranking in game)

Let imagine we have a game with 4 players. And after playing game, we will get ranking of 4 players based on their score, rank 1 is the best, rank 4 is the worst. I have created a model for predicting ...
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If $X\in\{0,1\}$, then $\frac{cov(X,Y)}{Var(X)}=\mathbb{E}(Y|X=1)-\mathbb{E}(Y|X=0)$

If $X\in\{0,1\}$, then $\frac{cov(X,Y)}{Var(X)}=\mathbb{E}(Y|X=1)-\mathbb{E}(Y|X=0)$ I have no idea what to address with the conditional expectation part. Thank you for any comments, someone has ...
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Precise Definition of $\mathbb{E}[X\mid \sigma(A)]$ Conditional Expectation of Random Variable given Sigma Algebra generated by a set

I want to define precisely, exhaustively and constructively the conditional expectation of a random variable given the sigma algebra generated by a set. This question has some discussion on it but the ...
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Conditional Expectation of Random Variable given an event

Suppose $(\Omega, \mathcal{H}, \mathbb{P})$ is a probability space, $(\mathsf{E}, \mathcal{E})$ a measurable space and $X:\Omega\to \mathsf{E}$ a random variable with well-defined expectation $\mathbb{...
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Exploring Alternative Pythagorean Expectation Formulae

In a regular match, a team score $a_{i}$ goals and allow $b_{j}$ goals, both are natural numbers). The result will depend on: $$\left | a- b \right |\left\{\begin{matrix} \geq e. 2.0\,{\rm points}\,{\...
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Finding the maximum likelihood solution corresponds to finding the root of a regression function. How?

Given a pair of RVs $z,\theta$ governed by a joint distribution $p(z,\theta)$. Conditional expectation of $z$ given $\theta$ defines a deterministic function (called as regression functions) $f(\theta)...
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Does the expectation of a quantile equal to the quantile of expectations?

Let $X$ be a random variable with finite expectation $E(X)$, and let's denote $X_{90}$ the 90% quantile of its distribution, meaning: $$P(X<X_{90})=0.9$$ Now, let Y be another random variable and ...
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Comparative statics for conditional expectations

Let $f\left(x,y\right)\in\left[0,\frac{1}{2}\right)$ a function such that $\frac{\partial f}{\partial x}>0$, $\frac{\partial f}{\partial y}<0$, and $\frac{\partial^2 f}{\partial x \partial y}<...
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What is the cubic expectation (third-order moment) of a complex gaussian vector (say, E[$aa^{T}a$])?

Note: I also posted this question on MATHEMATICS. For a real gaussian vector, an explicit formula for the cubic expectation can be found in Matrix Reference Manual (search 'Cubic Expectations' in this ...
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In a time series $x_t, x_{t-1},...,$, why is $E[x_t|x_t, x_{t-1},...]= x_t$?

Sorry if obvious but in a time series $x_t, x_{t-1},...,$, why is $E[x_t|x_t, x_{t-1},...]= x_t$? I don't really get what the random variable $x_t|x_t, x_{t-1},...$ represents? What I find ...
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Notation of expectation with conditional in subscript

Inside the book "The Elements of statistical learning", I stumbled upon the following notation (Ex. 2.7) $$E_{\mathcal{Y|X}}(f(x_0) - \hat{f}(x_0))^2$$ where $\mathcal{X, Y}$ are two random ...
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Simplify E[E[Y|X]|Z=z] in terms of E[Y|Z=z]

Let $Y, X, Z$ be random variables. It holds that $E[E[Y|X,Z]|Z] = E[Y|Z]$. See e.g. here Theorem 1 viii But I am particularly interested in the expression $E[E[Y|X]|Z]$. Is there a way to relate it to ...
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What's wrong with this conditional probability working?

A past exam paper for my course (BSc Mathematics, second-year module in statistical inference and modelling, unpublished) has a question, Let $(X,Y)^T$ be a bivariate random variable with joint ...
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Conditional expectation calculation with a binary variable

Suppose $X=\{0, 1\}$ is a binary random variable, $X$ and $Y$ are independent, $Z$ is another random variable. I get \begin{equation} E\left(\frac{X}{E(X)}Z|Y, X\right)=\frac{X}{E(X)}E(Z|Y, X). \qquad\...
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Expected Prediction Error for 0-1 Loss Function

In ESL on pages 20 and 21, we have a derivation of expected prediction error of a classification rule $\hat{G}(X)$: $$ EPE(\hat{G}) = E_X\sum_{k=1}^{K}L[\mathcal{G}_k, \hat{G}(X)]P(\mathcal{G}_k|X) $$ ...
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Exponential random variable X with a uniform random variable as its parameter

$$X\ \sim Exp(U) ~ and\ U\ \sim U(0,1) $$ The question asked for the value of $ P(X\geqslant 1)$ I saw the solution and it went like this: $$P(X\geqslant 1) = E[P(X\geqslant 1)|U] = E[e^{-u}] = \int_{...
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Loss Size Index Function of A Gamma Random Variable

I'm trying to prove that the loss size index function of a Random Variable Y, which is distributed as a Gamma Random Variable ($Y \sim Γ(γ,c)$) has the following expression: $$ I(y) = \frac{\textit{G}(...
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Showing that $E[\hat{\tau}_D] = P(n_D > 0)\tau_D$ and $\vert E[\hat{\tau}_D] - \tau_D\vert \leq \tau_D(1-\frac{N_D}{N})^n$

Consider the following double sampling scheme: We have a population of size $N$ with variable of interest $y_i$ for each $i \in \{1,\dots,N\}$, and (fixed) subpopulation $D$ of size $N_D$. Let $S$ ...
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Linear Regression : why expected value of response

I am really confused... why linear regression is modelling the expected value of response(or conditional expected value)? If we don't use mean square error as the loss function to minimise, is it ...
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Condition expectation of $X_1|\bar{X}$ [duplicate]

I've just learnt about conditional expectations and I'm confused about how we evaluate $E[X_1|\bar{X}]$ where $X_i\sim N(\theta,1),1\le i\le n$ and hence, $\bar{X}\sim N(\theta,1/n)$. Can someone ...
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Conditional expectation conditioned on an Indicator variable

Suppose I have a random variable $u$ that is standard uniformly distributed. And I have an indicator variable $S_{i}=1\left(V_{i}>0.5\right)$. Now I am interested in the following conditional ...
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Is there a function $g(X,Y)$ such that $E[g(X,Y)|X,Z]$ is constant in $X$?

More precisely, consider three random variables $X,Y,Z$, with $f(Y|X,Z)$ being the density of $Y$ conditional on $X,Z$. Say that this density is "regular" enough (e.g., continuous in all ...
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Identification of Richer Conditional Expectation from Less Rich Ones

Let $X_1,X_2$ be uniformly distributed and suppose that we know the quantities $E[Y | c_1 X_1 + c_2 X_2 = v]$ for all values of $c_1,c_2,v$ (i.e. we know all of the expectation of $Y$ conditional on ...
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Integral of conditional expectation over an event B

Could someone help me understand this equality? Let $\xi$ be a random variable. $\int_B(\frac{1}{P(B)}\int_B\xi dP)dP=\int_B \xi dP$ for any event $B$. How do we go from the integral over an event $B$ ...
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How to apply the conditional expectation of the multivariate normal distribution to fill gaps in data?

I have a data matrix $X \in \mathbb{R}^{m \times 4}$, where $m$ is any number of rows, whose data follow a multivariate normal (MVN) distribution. Suppose that for a given row $i$, the data for the ...
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Issue with Casella&Berger derivation of EM likelihood equality

In the explanation of the EM (Expectation maximization) algorithm p.328 in the book "Statistical inference" by G. Casella and R. Berger, 2nd edition, they present the following: $\mathbf{Y} =...

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