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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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Conditional mean and co-variance in $VAR(p)$ conditional on one lag only

Suppose I have a $p$'th order vector auto regression $$\vec Z_t = F_1\vec Z_{t-1}+F_2\vec Z_{t-2} + \cdots +F_p \vec Z_{t - p} + \vec \epsilon_t,\qquad \vec\epsilon_t\sim N_q(\vec0,Q)$$ where $...
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Understand conditional expectation w.r.t. sigma -algebra [duplicate]

When a random variable $X$ is discrete, the definition of conditional expectation of $X$ with respect to a decomposition $\mathscr D$ is $$ E[X|\mathscr D] = \sum_{i = 1}^m x_i \sum_{j = 1}^n P(X|\...
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35 views

How to deal with expected value in the context of time series?

For example, in this MA(2) model, $y_t = u_t + \phi u_{t-2}$ $u_t$ is identically, independently, normally distributed with a mean of 0 and a variance of $\sigma^2$. (Does variance matter here?) I ...
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Questions relating the definition of conditional expectation $E[g(X)|M]$ where is $X$ is random variable and M is an event

I saw the following definition of conditional expectation from a book: if M is event and X is continuous random variable then we define: $$E[X|M]=\int_{-\infty}^\infty xf(x|M)dx$$ Which is the ...
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113 views

Estimate E(Y|X,Z) from E(Y|X) and E[Y|Z]

Can I estimate $E[Y|X, Z]$ if I know $E[Y|X]$ and $E[Y|Z]$? As an example, let's say I have a model where $X\sim N(0, 1)$, $Y = aX+\epsilon_Y, \epsilon_Y \sim N(0, 1)$, $Z = bY+\epsilon_z, \...
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Calculate target variance contionaly to continuous variable

I'm trying to get the conditional variance of my target Y conditional to my inputs X. To do this I first discretized my inputs variable into 1 qualitativ variable and then I computed the variance of Y ...
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Generate Data where outcome is conditional on independent variables [closed]

I want to generate a synthetic dataset {Y, X1, X2}. Independent random variables X1 and X2 follow bernouli distribution where probabilities for X1 and X2 are known. Whereas, outcome variable Y needs ...
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Prove convergence in distribution, probability, or quadratic mean for a sequence of binary variables that depend on another binary variable

Suppose that $X$ has the support set $\{1, -1\}$, and $P(X = 1) = P(X = -1) = 0.5$. Suppose that $X_n$ has the support set $\{X, e^n\}$, and $P(X_n = X) = 1 - \frac{1}{n}$ $P(X_n = e^n) = \frac{1}...
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Proving covariance equals zero given a specific conditional expectation

I'm trying to prove the following: Given $𝐸[𝑋|𝑌 = 𝛽] = 𝐸[𝑋]$ for any value of $\beta$, prove that $\operatorname{Cov}(𝑋,𝑌) = 0$; So I was thinking to start with the definition of $\...
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67 views

Expected value of quotient of Poisson distributions

Let $X$ and $Y$ be independent random variables such that $X \sim \text{Poisson}(\lambda \cdot c)$ and $Y \sim \text{Poisson}(\lambda \cdot (1-c))$, where $c$ is a real number in $[0, 1]$. Is there ...
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Why can you not bypass the strong ignorability/unconfoundness assumption via iterated expectations?

Suppose we have that $\left(Y(1), Y(0)\right)$ are potential outcomes with $X$ being the covariate and $Z$ the treatment assignment. Typically in causal inference, one will assume strong ignorability ...
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1answer
53 views

Show $E[ (Y - E(Y|X)) (E(Y|X) - h(X))] = 0$

Show that $E[ (Y - E(Y|X)) (E(Y|X) - h(X))] = 0,$ where $X, Y$ are random variables with constant means and $h(x)$ is an arbitrary function. So far, I have expanded out the expectation and used ...
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If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$

Question If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$. Attempt: Please check if the below is correct. Let ...
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1answer
125 views

Conditional Expectation of pdf

Wish to identify what I'm doing wrong when finding the $\operatorname E(X\mid Y=5)$ of the following: $$f(x, y)=\begin{cases} 1/6 & \text{if } 0<x<2, 0<y<6-3x \\ 0 & \text{...
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Clarifying a proof of a particular paper on Steins Estimator

I am trying proving result (5.4) of the following paper. Its a paper on Steins estimator on spherically symmetric cases. The doubt is a s follows: Given $$X|\theta\sim \mathcal{N}(\theta,I)$$ ...
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38 views

Linear Prediction and Linearity of CEF

I am revisiting the basic notions of linear regression and stumbled upon the following idea in Cameron and Trivedi's Microeconometrics book: However, for the conditional mean to be linear in x, so ...
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1answer
22 views

Expectation of residuals in Zero Intercept Model

We know that the summation of residuals in a regression through origin model is not necessarily 0. Does that imply that Expectation of Residuals is not necessarily 0? CLRM still holds, so should that ...
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What does an expectation with respect to a policy mean in the reinforcement learning value function

I would like to know what the formal definition of the following expression is $$ V_\pi(s) = \mathbb{E}_{\pi}(G_{t+1} | S_t =s) $$ What does it mean to have the policy in the subscript? How would I ...
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Is the parameter vector of an indentifiable distribution of a transformed random vector always a subvector…?

I would like, after further considerations about this problem, to reformulate this question of mine again. I kept a record of the past words and remarks as the appendix below. I think the question ...
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1answer
71 views

How to calculate the integral of Normal CDF and Normal PDF?

I'm trying to find $\int_{\frac{a-b}{B}}^\infty\Phi\left(tA+ABx\right)\phi(x)\,dx$ where $A = \frac{\sqrt{\gamma_{3}+\sigma_3^2}}{\gamma_{3}},\ B = \frac{\gamma_{2}}{\sqrt{\gamma_{2}+\sigma_{2}^...
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When to use all 3 variables in a graph to estimate a conditional expectation of 2

The title might not be perfect. But here goes: Suppose there are 3 variables $(A,X,Y)$. And they have the following dependencies : $\Pr(Y,A,X)=\Pr(Y\mid A,X)\Pr(X\mid A)\Pr(A)$ $A \rightarrow (X,Y)$...
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1answer
38 views

Conditional expectation of Poisson process given number of events

Let $\{N(t), t\geq 0\}$ be a Poisson process with rate $\lambda$, $S_n$ the instant of the $n$-th arrival and $T_n$ the $n$-th interarrival time, that is, $T_n = S_n - S_{n-1}$, $n \geq 1$. Now ...
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1answer
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Simple Appplication of Law of Iterated Expectation

Consider a randomized experiment (AB test), where $n$ units are randomized into the treatment group $T_i=1$ and control group $T_i=0$. Let $M_i\in P$ denote the observed value of a continuous variable ...
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1answer
39 views

Integral from the Adversarial Spheres paper (maximum of the difference between a constant and a normal random variable)

I'm trying to follow a proof in the Adversarial Spheres preprint on arXiv. The proof requires the computation of the integral in Appendix F, page 14: $$\mathbf{E}\left[\max\left(\sqrt{2}\left(\frac{\...
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2answers
160 views

Conditional probability for consecutive Bernoulli trials

Independent trials, each of which is a success with probability $p$, are performed until there are $k$ consecutive successes. Let $N_k$ denote the number of necessary trials to obtain $k$ consecutive ...
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53 views

Expected value conditional on a function

Let $X$ and $Y$ be random variables. What is the relationship (if any) between $E[Y|X]$ and $E[Y|g(X)]$? I have been trying to Google or look in books but I'm having trouble even articulating this ...
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One-sided measurement error: $\widetilde{X} = X - \eta$, $\eta\geq0$. Is $E[X|\widetilde{X}=\widetilde{x}]$ increasing in $\widetilde{x}$?

Let $X\geq0$, $\eta\geq0$ and $X,\eta$ independent. We measure $X$ with a one-sided error: $\widetilde{X} = X - \eta$. Is $E[X|\widetilde{X}=\widetilde{x}]$ increasing in $\widetilde{x}$?
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The expectation of the $K^{th}$ raw moment of $X$ given $X$: $E(X^k|X)$

Intuitively I believe $E(X^k|X)=X^k$, $k$ is a non-negative integer. One obvious special case is when $k=1$. Anyone has an idea how to prove it?
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Do discriminative models model conditional expectation?

In Machine Learning classic models like MLP, Logistic Regression or Linear Regression are called discriminative models. I frequently read that those models estimate the conditional probability ...
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Taylor Series Expansion of Unconditional Expectation

We know that the best 1st order approximation of an unconditional expectation is the following- $$E(y|x)=(E(y)-\beta E(x))+\beta x$$ where $\beta=\frac{\operatorname{Cov}(y,x)}{\operatorname{Var}(x)}...
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Conditional expectation of RVs under truncation-based stochastic signal release?

Suppose $z\sim N(0,1)$ and $x\sim N(\gamma z, 1)$, where $\gamma$ is a known sensitivity parameter. That is, $x=z+\epsilon, \epsilon \sim N(0,1)$ where $\epsilon$ is exogenous. Further, suppose there'...
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Conditional expectation on an estimator for defensive sampling

In Introducing Monte Carlo Methods, by Robert and Casella, we have How do we derive the second equality? Shouldn't it be $$E\left[\frac{f(X_i)}{g_{Y_i}(X_i)}|X_i\right]=\frac{f(X_i)}{g_1(X_i)}\rho+...
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1answer
47 views

Average treatment effect in binary choice model

All the random variables below are defined on the same probabiluty space $(\Omega, \mathcal{F}, \mathbb{P})$. Consider the following model $$ Y\equiv 1\{\epsilon > \beta_0+\beta_1X\} $$ where $1$...
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Are these two ways of looking at conditional linear regression the same?

Let $X$, $\hat{S}$, $\hat{V}$ be random variables, with means (and conditional means, if necessary) $=0$. Is $\beta^*(X) = \frac{\mathrm{Cov}(\hat{S}, \hat{V}|X)}{\mathrm{Var}(\hat{S}|X)}$ the ...
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Does this equality hold?

Is it true that $E[XY|Z]=E[X|Z]E[Y|Z]$ if $X$ and $Y$ are independent each other, but $X$, $Y$ are not independent with $Z$? Can anyone prove this? Thank you.
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Missing steps in Bellman Equation and MDP assumptions

My question is similar to this one. I am trying to fill in the missing steps of the Bellman equation for a Markov Decision Process: I will focus on the first term of the sum $R_{t+1} + \gamma \sum_{k=...
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Compare health spending per capita of 5 countries conditional on their age structure, health index and GDP per capita

I am interested to answer the following question: How does the health spending per capita in 2010 of 5 countries different if they have the same age structure in their population, health index and GDP ...
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39 views

Conditional distribution relations

There is a probability density function of the form, $f_S(s)=\displaystyle\iint f_S(s|x,y)f_{X,Y}(x,y)dxdy$ that is used for evaluation of expectation of some monotonic function $\mathbb{E}[g(S)]=\...
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Conditional expectation given event and random variable

Let $W$ be a random variable that only takes on the values $1$ or $0$. Let $X$ and $Y$ be two other random variables. I came across the following: $$\mathbb{E}(Y|W=1, X)$$ How is this 'conditional ...
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Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers

Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers $Y_1, Y_2, ...$ from the uniform distribution on $[0, 1]$, until ...
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1answer
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Calculating conditional expectation and mean time to failure

I was reading text on probability where they state: $$\operatorname{Ex}[C]=\sum_{i=1}^{\infty}i\cdot\Pr[C=i]=\sum_{i=0}^{\infty}\Pr[C>i]$$ Now assuming there is a system which fails at each step ...
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2answers
212 views

UMVUE for Bernoulli

Let $X_1,..,X_n$ be independent and $Bin(1,\theta)$ distributed. I would like to find the UMVUE for $\phi(\theta)=\theta^3$. I have a complete and sufficient statistic in $T=\sum_iX_i$, and a unbiased ...
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Conditional Expected Value of Birnbaum-Saunders Bivariate

Consider $(T{1},T_{2})\sim BVBS(\alpha_{1},\beta_{1},\alpha_{2},\beta_{2}).$ According to item (b) of Theorem 3.1 of Balakrishnan and Kundu (2010) article, we have: \begin{align*} f_{T_{1}|T_{2}=t_{2}...
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31 views

Construct joint distribution of $X,Y$ such that $E[X|Y=y,y\geq \bar{y}]$ is piecewise linear

Can one construct a joint density $f(x,y)$ such that the marginal distribution of $Y\sim~U[c,d]$, no restrictions on $X$ (it would be great that $X$ also has uniform distribution) as long as it has ...
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1answer
103 views

How to improve an estimator for a Poisson sample

Given the statistical model $(\mathbb N_0^n, P(\mathbb N_0^n),\operatorname{Poi}(\vartheta)^{\otimes n}:\vartheta >0)$, $T(X)=X_1X_2$ is an unbiased estimator of $\vartheta^2$. I want to improve ...
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0answers
20 views

Standard errors of OLS estimate if regressor is a stochast?

Assume the model classical linear regression model (with for simplicity only one regressor) $$y=X\beta +u,$$ with $u$, $X$ independent, and $\operatorname{Var}(u|X)=\sigma^2I_n$. Assume for ...
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2answers
118 views

Expected value of $X$ which follows a normal distribution, between a certain interval [duplicate]

What is the process of finding the expected value of $X$ in a normal distribution between a certain interval? In particular I want to find: $E(X | a \le X \le b)$. For example, if $X$ has $\mu=0$ ...
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153 views

Multiplicative or additive error model in linear regression

Suppose that $Y$ and $X$ have an unknown joint distribution with an arbitrarly complicated unknown C.E.F. $ \; E[Y|X]=\mu (X)$. Suppose that we want to find the best (in MSE minimization terms) ...
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9 views

accuracy conditional on feature values

I have a binary classification model and I would like to estimate the accuracy as a function of another variable. To be clearer, I can compute the usual accuracy on the testset: $$ acc = \frac{\...
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1answer
33 views

Conditional expectation and variable decomposition

Suppose that $X$ and $Y$ have an uknown joint distribution $f_{XY}$. How can I formally demostrate that it always exists a unique decomposition of the form : $$ Y = E[Y|X] +\epsilon $$ without ...