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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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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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 ...
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 $E[x \lvert ... 1answer 37 views What is the distribution of conditional expectation of a function f(X) of the random variable X? i.e. E(f(X)|X) I have a continuous random variable X with a known PDF. I want to find the distribution of f(X) where ... 1answer 73 views Finding$E(X\mid X>Y)$when$X,Y$are i.i.d$U(0,1)$I am unable to compute conditional probability(x|x>y) in the above question. Also, I am unable to determine the region of integration for calculation of the above expectation. 3answers 68 views Question Regarding Zero Conditional Mean Hi I am a beginner to econometrics! I have been dealing with bivariate regression. We use the formula$y = \beta_0 + \beta_1 x$. I am told that if$E(u\mid x) \ne 0$then the estimate of the slope ... 0answers 16 views Middle entries of a random vector - Conditional expectation and covariance matrix of normal distribution P(X2|X1, X3) [duplicate] Let us consider the random vector$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)$. What I am trying to compute is the ... 1answer 43 views A very basic question on ENDOGENEITY In the regression model$Y$=$\beta_0$+$\beta_1X_1$+$\beta_2X_2$+.......+$\beta_kX_k$+$\epsilon$where$\epsilon$=$\delta_0X_2$+$\lambda$Will this also be the case of endogeneity ... 1answer 35 views Conditional expectation setup Suppose$\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) \... 1answer 91 views 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 ... 2answers 201 views 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 ... 1answer 53 views 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 ... 2answers 52 views 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}... 1answer 254 views 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 processy_i =\... 0answers 17 views 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. ... 0answers 55 views What is the marginal variance of the mean ofy$if I only know the variance of the conditional mean of$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$. The ... 1answer 133 views Are the law of iterated expectation and the law of total expectations the same? On the Wikipedia page of the Law of total expectations it is said that The proposition in probability theory known as the law of total expectation, the law of iterated expectations, the tower rule, ... 0answers 19 views Question regarding conditional expectation [duplicate] In Larry Wasseman's lecture notes(lecture 4, page 4) I found this statement$\mathbb{E}[Y|X=x] = \sum_y y f_{Y|X}(y|x)$or$=\int_y y f_{Y|X}(y|x)dy.$An important point about the conditional ... 2answers 61 views Conditional expectation function Consider the standard linear regression model given by$Y = XB + \varepsilon$.$E[Y\mid X] = XB$if$E[\varepsilon \mid X] = 0$. We say that the conditional expectation function is a random ... 2answers 149 views Different solution for a probability question I got the following problem: Find the probability that for two arbitrary numbers$x$and$y$with$x,y \in [0,1]$they satisfy$x+y<1$and$xy<\frac1{10}$. In short words the sum of the two ... 0answers 31 views How to find conditional expectation$(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}^...
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 ...