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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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Calculating the expected value of truncated normal

Using the mills ratio result, let $X \sim N(\mu, \sigma^2)$, then $E(X| X<\alpha) = \mu - \sigma\frac{\phi(\frac{a- \mu}{\sigma})}{\Phi(\frac{a-\mu}{\sigma})}$ However, when calculating it in R. ...
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Expected function evaluation of random variable w.r.t. different distribution

Suppose I have two continuous random variables on the same domain, $\xi \sim \mathbb{P}, \xi' \sim \mathbb{Q}, \in \Xi$ and joint probability $(\xi, \xi') \sim \Pi \in \Xi^2$ . Now I would like to ...
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Iterated expectations and variances examples

Suppose we generate a random variable $X$ in the following way. First we flip a fair coin. If the coin is heads, take $X$ to have a $Unif(0,1)$ distribution. If the coin is tails, take $X$ to have a $...
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How does correlated relationship implies non zero conditional mean?

i.e. if $cov(X,Y)\neq0$, how do we derive $E(X|Y)\neq0$ it seems that it is quite obvious and usually be taken into granted, but I just couldn't figure it out in a short time, can somebody give me a ...
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where can find good and in depth explanation for expectation, manipulation of expectation, sampling instead of expectation?

I look for a book or online source, for better understanding the expectation, expectation inside the expectation or sampling for calculation of expectation. for example, in Richard S. Sutton's ...
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29 views

Conditional covariance of multivariate normal tail

Let $X\sim N(\mu,\Sigma)$, $t\in\mathbb{R}$, and $a$ be a non-zero vector of the same dimension as $X$. Define a random vector $Y=X\mathbb{1}(a^\top X\ge t)$, where $\mathbb{1}$ denotes the indicator ...
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47 views

Conditional Expectation of Log-Normal Distribution

I want to evaluate a conditional expectation of log-normal distribution. Let $y$ be a log-normal distributed random variable. So $\log(y)\sim N(\mu,\sigma^2)$. I want to calculate $E[y-1|y-1>0].$ ...
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combining variables from different tables

I'm having trouble combining variables from differents tables. In this case, I have the exogenous variables $X$ (categorical) and the endogenous variable $Y$ (continuous). None of these variables are ...
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Conditional expectation; how to find E[xy] when E[x|y] is known?

In my studies for an exam which I have on Friday I have come across this assignment from last year in which the following question is asked: "Let $E[x] = \mu$ and $var[x] = \sigma^2$. If $E[x \lvert ...
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What is the distribution of conditional expectation of a function f(X) of the random variable X? i.e. E(f(X)|X)

I have a continuous random variable X with a known PDF. I want to find the distribution of f(X) where ...
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1answer
52 views

Finding $E(X\mid X>Y)$ when $X,Y$ are i.i.d $U(0,1)$

I am unable to compute conditional probability(x|x>y) in the above question. Also, I am unable to determine the region of integration for calculation of the above expectation.
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Question Regarding Zero Conditional Mean

Hi I am a beginner to econometrics! I have been dealing with bivariate regression. We use the formula $y = \beta_0 + \beta_1 x$. I am told that if $E(u\mid x) \ne 0$ then the estimate of the slope ...
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Middle entries of a random vector - Conditional expectation and covariance matrix of normal distribution P(X2|X1, X3) [duplicate]

Let us consider the random vector $X=[X_1,X_2,X_3]$, which follows a multivariate normal distribution. That is, for each entry $X_i$: $X_i \sim N(\mu_i, \Sigma_i)$. What I am trying to compute is the ...
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A very basic question on ENDOGENEITY

In the regression model $Y$ = $\beta_0$ + $\beta_1X_1$ + $\beta_2X_2$ +.......+ $\beta_kX_k$ + $\epsilon$ where $\epsilon$ = $\delta_0X_2$ + $\lambda$ Will this also be the case of endogeneity ...
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Conditional expectation setup

Suppose $\epsilon_0,\epsilon_1$ are iid random variables with density $f$ and cdf F and $c\in R$. Then why is: $$E[\epsilon_1|\epsilon_1+c>\epsilon_0]= \frac{\int_{-\infty}^{+\infty} x F(x+c) f(x) \...
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Law of Iterated Expectations Example

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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Conditional expectation of uniform random variable given order statistics

Assume X = $(X_1, ..., X_n)$ ~ $U(\theta, 2\theta)$, where $\theta \in \Bbb{R}^+$. How does one calculate the conditional expectation of $E[X_1|X_{(1)},X_{(n)}]$, where $X_{(1)}$ and $X_{(n)}$ are ...
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Conditional expectation of two independent RV

The expectation of the product of two independent random variables $X$ and $Y$ is the product of the expectations: \begin{align} E(XY) = E(X)E(Y) \end{align} Let's add another random variable $Z$ in ...
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2answers
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Is this formula for the Law of Iterated Expectations correct?

I saw two versions of the law of iterated expectations, this one: \begin{align} E(E(Y\vert X)) = E(Y) \end{align} and this one: \begin{align} E(E(Y\vert X_1, X_2)\vert X_1) = E(Y \vert X_1) \end{align}...
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Can I use averages to improve the forecast of a multiple regression?

I have a cross-sectional multiple regression that I have estimated and now I would like to apply it to make a simple forecast of the dependent variable. Take the data generating process $$y_i =\...
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Conditional Expectation for Large Population

This was another question on my past exam and I'm curious to kmow how to solve it. Let there be a population of 100,000 and the chance of someone being infected with a certain disease is 0.000001. ...
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What is the marginal variance of the mean of $y$ if I only know the variance of the conditional mean of $y$?

Suppose an estimate for the conditional mean of $y$ given $x$ is $\hat{E}(y|x)$. Suppose the variance (or the variance estimate) of $\hat{E}(y|x)$ is known to be $V(\hat{E}(y|x))$ for all $x$. The ...
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Are the law of iterated expectation and the law of total expectations the same?

On the Wikipedia page of the Law of total expectations it is said that The proposition in probability theory known as the law of total expectation, the law of iterated expectations, the tower rule, ...
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Question regarding conditional expectation [duplicate]

In Larry Wasseman's lecture notes(lecture 4, page 4) I found this statement $\mathbb{E}[Y|X=x] = \sum_y y f_{Y|X}(y|x)$ or $=\int_y y f_{Y|X}(y|x)dy.$ An important point about the conditional ...
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Conditional expectation function

Consider the standard linear regression model given by $Y = XB + \varepsilon$. $E[Y\mid X] = XB$ if $E[\varepsilon \mid X] = 0$. We say that the conditional expectation function is a random ...
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2answers
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Different solution for a probability question

I got the following problem: Find the probability that for two arbitrary numbers $x$ and $y$ with $x,y \in [0,1]$ they satisfy $x+y<1$ and $xy<\frac1{10}$. In short words the sum of the two ...
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How to find conditional expectation$(X_1+X_2)^2$ given $X_1 = X_2$?

How do I show that $E[(X_1 + X_2)^2|X_1=X_2] = 2\sigma^2 + 4\mu^2$. When $X_1$ and $X_2$ follows $N(\mu,\sigma^2)$ independently. As $X_1 = X_2$ is given, then I suppose I only need to find $E[4{X_1}^...
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1answer
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Conditional-expectation operator inside of expectation operator

Let $b(\theta)$ be a parametric function, let $U$ be a sufficient statistic for $\theta$, let $T$ be an unbiased estimator for $b(\theta)$, and denote $g(U)$ as $g(u)=E[T|U=u]$. I am told that the ...
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Conditional Probability and Expectation for Poisson Process

To solve part (a) I have $P(X_2 = k\mid X_1 = 1)= \dfrac{P(X_2 = k \cap X_1 = 1)}{P(X_1 = 1)} = \dfrac{e^{-2}}{e^{-1}}=e^{-1}$. Then for part (b), for simplicity, I let $X_2=X$ and $X_1=Y$, then $$E(...
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Why is the conditional expectation the best predictor but only if we have the joint distribution?

If we want to predict one variable $Y$ based on another $X$, the best predictor is apparently $\mathbb{E}[Y \mid X = x]$. However, this apparently assumes two things: The distribution is symmetric. ...
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Conditional probability of Negative Binomial R.V. given the SUM of its values

Suppose $\{z_{ij}\}$ are independent Negative Binomial random variables with means $\{\mu_{ij}\}$, with $i=1\dots I$ and $j=1\dots J$. How do you find the (expectation of) conditional probability ...
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Expectation conditional on self and others

I would simply like to know if: $E[x_1|x_1,x_2]=E[x_1|x_2]$ or $E[x_1|x_1,x_2]=E[x_1|x_1]=x_1$ or something completely different and why. This is not homework. It came up because I'm trying to ...
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Expectation of standard exponential squared given sum of two standard exponentials

So I have been working on this question for a while and made some progress , but I run into a problem about the normalizing constant. The question is, for $X$ and $Y$ i.i.d. standard exponential, find ...
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1answer
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Why does $\operatorname E(\varepsilon\mid x) = 0 \implies \operatorname{cov}(\varepsilon,x) = 0$?

I understand the intuition behind the question but I'm trying to prove it to myself with math.
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Expected value as an orthogonal projection

I'm reading a paper in which the expected value of a random variable, $\mathbb{E}[X]$, is characterized as an orthogonal projection. This is on page 10. I've seen the geometric interpretation of ...
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1answer
49 views

Conditional expectation of a vector

Suppose we have two random vectors $X=(X_1,X_2)^T$ and $Y=(Y_1,\dots,Y_n)^T$. I wish to find a simple definition or formula for $$ E_{X|Y=y}[X] $$ Intuitively, I think the following is correct: $$ ...
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1answer
37 views

Analytical solution to the covariance between a continuous and a categorical variable

Let $X$ be a continuous variable with mean $\mu$ and $Y$ be a categorical variable with event probability vector $\mathbf{p}$. I am trying to calculate $\operatorname{Cov}(X, Y)$. I have the solution ...
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1answer
121 views

Multivariate Normal : expectation of X given Y is doubly-truncated

Let $(X, Y)$ be distributed as a multivariate normal with parameters $$ \mu = \begin{bmatrix} \mu_X \\ \mu_Y \end{bmatrix} \qquad \Sigma = \begin{bmatrix} \sigma_X^2 & \sigma_{XY} \\ \sigma_{XY} &...
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1answer
50 views

Independence and conditional distribution

In a problem that I'm solving I find that: "Let data (yi,xi) be sampled randomly from a two-dimensional distribution such that y|x is N(ɑ,x^2σ^2)". Are y and x i.i.d? maybe just identically ...
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1answer
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Difference between averaging and ignoring the partial dependencies?

This question sparks from model interpretation/visualization. To graph the dependency of a function with >2 arguments, one often needs to ignore or average out some arguments. Problem set Hastie, ...
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1answer
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Proof of contemporaneous exogeneity, and its implications for an AR(1) model

It can be shown by contradiction that exogeneity fails to hold for an AR(1) model. Is there any proof that contemporaneous exogeneity does not fail to hold? All I've come across is assuming it does ...
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1answer
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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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1answer
54 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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1answer
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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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2answers
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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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1answer
37 views

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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0answers
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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 $\...