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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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How do I calculate the standard deviation of a mean value for a joint distribution? [on hold]

Say I have a function $f(\mathbf{x})$ where $\mathbf{x}$ is a vector of 0s and 1s. Each of these 0s or 1s, $x_i$, comes from a probability distribution $p(x_i | x_1, \dots, x_{i-1})$. When I sample ...
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Consistent estimator for conditional expectation

Take sequence of random vectors $(Y_i, X_i)_{i=1}^N$ i.i.d. $X_i$ has finite support. Let $x$ be a point in the support of $X_i$. Consider $E(Y_i|X_i=x)$. Suppose it exists and is finite. Is it ...
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Practical Examples: Expectation of a function with respect to a probability

I have encountered the following phrasing while reading Bishop's "Pattern Recognition and Machine Learning": Although for some applications the posterior distribution over unobserved variables will ...
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Implications of conditional mean independence

I have three random variables $Y,X,W$ with supports $\mathcal{X}, \mathcal{Y},\mathcal{W}$, respectively. I assume $E(Y|X,W)=0$ almost surely. Take two functions $z: \mathcal{X}\rightarrow \mathbb{...
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Does $E(X \mid Y,Z)=0$ imply $E(X \mid Y)=0$?

Does $E(X \mid Y,Z)=0$ imply $E(X \mid Y)=0$? In other words, if we have $E(X \mid Y,Z)=0$ then can we also say $E(X \mid Y)=0$?
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Application of law of iterated expectations

I would like your help to show a statement that uses the law of iterated expectations. In my notation $Supp_X$ denotes the support of a random variable $X$. Consider the random variables $\epsilon,...
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Error variable correlated with explanatory variables

We know that sometimes the error variable in a regression framework may be correlated with explanatory variables; this happens e.g. when we omit an important predictor from our study, getting the well ...
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Estimating conditional expectation using SVR

I have an estimation problem: $X_t=f(S_t)=E(X_T\mid S_t)$ given the observations $(X^i_{T},S^i_{t})$ for all $i$'s. I used support vector regression (SVR) to run a regression of $X_T$ on $S_t$. Here $...
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Is this equation true for any joint probability distribution (used in orthogonality principle proof for estimators)?

Given two random variables $x, y$, is it true that $p(x - \hat{x}|y) = p(x|y) - \hat{x}$ where $\hat{x}$ is a known constant I came across this in this kalman filter derivation, Corollary 3.2.1, ...
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Interpretation the law of total expectation

For a function of two RVs we have: $$ E[g(X,Y)] = E[\underbrace {E\{ g(X,Y)|Y\} }_{It\space is\space a \space function \space of\space Y}] = E[h(Y)] $$. For each $Y=y$, the inner expectation is the ...
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E(XY)=E(XE(Y|X) Joint Conditional Expectations [duplicate]

Why is E(XY)=E(XE(Y|X))? Is this using the properties of conditional expectation and is there a general formula that can be applied when you have E(...)=E(..(E(Y|X))?
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law of iterated expectations doubt

Is it correct to do so: $E(X^{-1} Y) = E_x(E(X^{-1} Y \vert X) = E_x(X^{-1} E(Y \vert X)) $ Are we allowed to use LIE in case of non linear functions like $X^{-1} $
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Calculating Conditional Expected Value in R

I have a stock whose returns follows a Random Walk with mu= 6% and sigma= 20% I would like to calculate the 10 returns of this distribution that I would get ...
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Derive linear regression model from the conditional distribution of Y|X

Suppose that $Y|X=x \sim N(\mu_Y + \frac{\sigma_Y(x-\mu_X)\rho}{\sigma_X}, \sigma_Y^2(1-\rho^2))$. The question asks to specify a simple regression model under this conditional distribution. A ...
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Expectation of normal RV conditional on normal mixture

Let $v\sim\text{Normal}\left(\mu,\sigma_v^2\right)$ a random variable with $\mu>0$ and $u\sim\text{Normal}\left(0,\sigma_u^2\right)$ Let $k\sim\text{Binomial}\left(N,p\right)$ a random variable ...
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Estimating tail share of apparently subexponential distributions drawn from finite population, given a finite sample

Suppose I have data on a large sample of some units of observation, where the observed quantity has meaningful differences and ratios. The sample is much smaller than the population, but both are ...
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conditional and interventional expectation

Conditional expectation $E[Y|X]$ and interventional expectation $E[Y|do(X)]$ are related but conceptually very different things. I know that if $X$ is a randomly assigned by an experiment, we have ...
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Markov's inequality with conditional probability

How to apply Markov's inequality in case of conditional probability $P(X \ge a | Y \le a)$ where $X$ and $Y$ are not independent. Can we write $P(X \ge a | Y \le a) \le \frac{E(X | Y \le a)}{a}$?
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R result interpretation conditional inference tree result for nominal response

I have nominal responses, "yes/no/don't know", that I am using in a conditional inference tree in R. I am having trouble with how to interpret the model's output concerning one of the independent ...
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How to choose the control variables in the conditional expectation to hold fixed when studying a causal relationship

I'm reading the introductory chapter of the wooldridge's book, "Econometric analysis of cross section and panel data". The chapter begins by highlighting the role and importance of conditional ...
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Conditional Expectation of Poisson

Suppose $X_1$,$X_2$,$X_3$,.....,$X_n$ are i.i.d. random variables with a common pmf poisson(λ) (t = a value) How would you calculate the below without using intuition (I would appreciate if you ...
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How does forecast skill score change when seasonality in the forecast quantity is removed?

Given RMSE skill score $s$: \begin{equation}\label{eq:msess} s = 1-\frac{\text{RMSE}(f,x)}{\text{RMSE}(r,x)}, \end{equation} where $f$, $r$, and $x$ are forecasts of interest, reference forecasts, ...
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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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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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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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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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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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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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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 ...