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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18 views

Conditional distributions of correlated normal random variables

Suppose that $X$ and $Y$ are normally distributed with mean zero and nonzero covariance. I want to know the distributions of $X | X - Y > c$ and $Y | X - Y > c$, which I believe should be ...
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Obtaining the expected value $E[X_{(1)} \mid\overline X = c]$

Suppose we have $X_1,\dots, X_n \overset{\text{iid}}{\sim} N(\mu = 0, \sigma^2 = 1)$, for a known $n$. And we want to calculate $E[X_{(1)} \mid \overline X = c]$, where $c \in \mathbb{R}$ is known, $...
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Self Study: Trivariate Normal Expectation with Inequality Condition

I'm reading a paper and found an interesting expectation. I know the result the author found but I can't figure out the intermediary steps because the author provided none. My attempt is getting ...
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Order Statistics: How to calculate expected value of a function involving first and second order statistics

I am currently stuck with a challenging problem. I have n values drawn i.i.d. from a distribution F(x). Let $v_1$ be the nth order statistic (highest value) and let $v_2$ be the n-1 order statistic (...
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Hypothesis testing on one time-series conditional on another

I have two time-series, $A$ and $B$. I want to test if $A$ is lower when $B<0$. Is it theoretically correct to use, say Welch's t-test, to test if $E[A|B<0]$ is statistically significantly ...
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30 views

How to prove that the expected value minimizes mean square error [duplicate]

In this wiki subpage about conditional probability we read that if $(\Omega, \mathcal{F}, \mathcal{P})$ is a probability space and $X:\Omega\to\mathbb{R}$ is a random variable with mean and variance, ...
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Univariate Regression, expectation of x given y [duplicate]

Perhaps an easy question for some: Given I have a univariate regression setting... $$y = b_1*x + e$$ ... with $e$ being the normally distributed error (standard normal dist.). $x$ is also a random ...
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When is E(A*B) = E(A * E(B))?

I'm looking at the documentation for the econml python package. On this page it is stated that: \begin{split}E[Y_{i, t}^{IPS} | X, W] =~& E\left[\frac{Y_i 1\{...
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Expected value of balls left, drawing colored balls without replacement

In an urn, there are $m$ red balls and $n$ green balls. Every minute, you draw one from the urn. What is the expected number of balls (regardless of its color) left in the jar after you have drawn all ...
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How exactly does the Lehmann-Scheffè theorem directly imply the identity $E[S^2 \mid \bar{X}] = \bar{X}$?

Take the random sample $X_1, \dots, X_n$ with mean $\mu$ and variance $\sigma^2 < \infty$. Now assume the $X_i$ are Poisson random variables with parameter $\lambda$. I am told that the Lehmann-...
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Problem on calculating conditional expectation by law of total expectation

Let $X$ be a standard normal distribution and let $$ Y= \begin{cases}X-1 & , X\le 0 \\ X &, X>0 \end{cases}$$ Find mean and variance of $Y$. MY working $$E(Y)=E(X-1\mid X\le 0)P(X\le ...
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Can zero covariance and zero expectation imply zero conditional expectation?

Let $x$ and $\epsilon$ are two random variables. If $$Cov(x, \epsilon)=0$$ and $$E[\epsilon]=0,$$ can that lead to $E[\epsilon|x]=0?$
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Conditional Expectation of a normal distribution [duplicate]

say we have a multivariate normal distribution with ${\boldsymbol Y} \sim \mathcal{N}(\boldsymbol\mu, \Sigma)$ The conditional expection is $\overline{\boldsymbol\mu}=\boldsymbol\mu_1+\Sigma_{12}{\...
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Explanation for E[E[X|Y]|Y]=E[X|Y]

I would like to ask for the proof of $E[E[X|Y]|Y]=E[X|Y]$ Per my understanding (for discrete case): because $E[X|Y]=g(Y)$ hence, $E[E[X|Y]|Y] = E[g(Y)|Y]= \sum_y g(y)*p(y|y)=\sum_y g(y)=\sum E(X|Y=y)$ ...
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Equivalence sum of conditional expectations given random observations and sum of conditional expectation given order statistics

Suppose $X_1,...,X_n$ are independent and identically distributed random variables defined on some probability space $(\Omega, \mathcal{A}, P)$. Define $Y=\sum_{i=1}^{n}X_i$. If we denote the ...
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Most probable value given observation

Suppose I have observed $Z = 3$, where $Z = X + Y$, where $X \sim N(0,9), Y \sim N(0,4)$. How would I find the most probable value of $X$ that would have given me $Z = 3$? My attempt at a solution: ...
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Memorylessness of exponential and expectation

Suppose I have a teller who has servicing time that is exponential with mean of $2$ minutes. Say customer $A$ arrives at noon and begins being serviced by the teller. What is the expected length of ...
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3answers
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Conditional expectation given sum of weighted average

Suppose X, Y are i.i.d standard normal (mean 0, standard deviation 1) random variables, a, b, c are constant scalars. $$Z = a X + b Y$$ How to express $E[X|Z=c]$ using $a,b,c$?
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From conditional to unconditional expectation

Consider a random variable $Y$ and a random variable $G$. $G$ can only take value $1$ or $0$. Is it true that $$ E(Y|G=0)\geq 0 \Leftrightarrow E((1-G)Y) \geq 0 \quad ?$$ My thought is yes and below I ...
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estimation in linear models: loss functions and response variables

Consider a dataset of $N$ iid samples $\{(y_i,x_i)\}_{i=1}^N$ drawn from a joint distribution $(Y, X)$ where $x_i\in\mathbb{R}^p$ and $y_i\in\mathbb{R}$. In this setting, it is my understanding that a ...
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OLS as approximation for non-linear function

Assume a non-linear regression model \begin{align} \mathbb E[y \lvert x] &= m(x,\theta) \\ y &= m(x,\theta) + \varepsilon, \end{align} with $\varepsilon := y - m(x,\theta)$...
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Is there such a notion as mean residual lifetime at the time of failure?

The mean residual lifetime (MRL) is a well understood notion: If $X \sim f(\cdot), F(\cdot)$, then, the MRL at any time $t$ is: $m(t) = E[X-t | X > t]$. Are there well understood properties of $E[m(...
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Law of total expectation: how to relate $E(X) = E(E(X|Y))$ to $E(X) = \sum_i E(X|A_i)P(A_i)$?

Below is the definition of the law of total expectation from Wiki. The first equation states that for any $X, Y$ on the same probability space, then \begin{equation} E(X) = E(E(X|Y)) \end{equation} ...
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34 views

Law of iterated expectation for the square of a conditional expectation

We know from the law of iterated expectations that $$E[E[X|Y]] = E[X]$$ However, does the same hold true for the square of a conditional expectation? I.e. is the following expression true, $$E[E[X|Y]^...
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28 views

Unconditional and conditional models

I don't know if the question is worded weirdly, but I'm having difficulties understanding its logic. I have the solution, but if possible, can someone explain the reason behind it? We have two models (...
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151 views

Understanding the conditioning in a GARCH process

In a GARCH model like the following $$ \begin{aligned} y_t &= \sigma_tz_t,\\ \sigma_t^2 &=\omega(1-\alpha-\beta)+\alpha y_{t-1}^2+\beta \sigma_{t-1}^2 \end{aligned} $$ where $z_t$ is assumed ...
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What's a real life example of a case in which the conditional expectation and unconditional expectation differ?

My questions are We have one variable, is called, "a" and mean of "a" is 5664. is this unconditional mean ? When we regress b on a (dependent is a, independent is b) Mean Dependent Var is 5664 ...
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Subscript notation in expectations (variational autoencoder)

This is the objective function of a variational autoencoder. I am not sure how to interpret the second term. It appears to be an expectation value over log p(x^(i)|z), but I'm not sure what role the ...
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Conditional vs. Unconditional Maximum Likelihood

I have some questions on the difference between conditional MLE (CMLE) and unconditional MLE (UMLE) in practice. In what follows I will only talk about the unconditional and conditional mean and leave ...
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226 views

Rao Blackwell theorem on Bernoulli distribution

I need help with the following Problem: Let $X_1,...,X_n$ be a random sample of iid random variables, $X_i\sim Ber(p), p\in (0,1)$. We want to estimate $\theta = p^2$. It is known, that $\hat{\theta}(...
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Expectation of structural equation

I am trying to learn about structural equations, and in this post here Correlation, regression and causal modeling I am having difficulties trying to prove the answer. The problem is, given structural ...
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999 views

Estimating conditional variance y|x

I am building a predictor for $y = f(x)$ using training samples ${(x_i, y_i)}$ (assume) drawn i.i.d from some distribution $p(x,y)$, by optimising the empirical L2-loss: $f(x) = argmin_f \; \sum_i ||...
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Gauss Markov Theorem and zero conditional mean/mean independent assumption

So I read online that one of the assumptions of Gauss Markov Theorem is: $$E[\epsilon_i]=0$$However, we also know that one of the assumptions for linear regression is the zero conditional mean: $$E[\...
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1answer
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Iterated expectation over different sets of variables

For some random variables $Y, X, Z,$ and $Q$, can we simplify $E[E[Y|X,Z]|Z,Q]$? Is it correct that $E[E[Y|X,Z]|Z,Q] = E[Y|X,Q]$ (idea being that $Z$ is averaged out)? In general, for some sets of ...
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What is the maximum entropy distribution given *conditional* means and MADs?

I know the maximum entropy distribution given the mean and MAD (Mean Absolute Difference) around the mean (it's the Laplace distribution, a proof here for example). I also know the maximum entropy ...
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Poisson random variable self-study question

You are invited to a party. Suppose the times at which invitees arrives are independent uniform(0,1) random variables. Suppose that aside from yourself the number of other people who are invited is a ...
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Expected value of sum of two gaussian random variables conditional on their difference

Given two standard normally distributed random variables $x_1$ and $x_2$. $y = x_1 + x_2$ I would now like to calculate the following: $$\mathop{\mathbb{E}}[y | x_1 -x_2 = 0]$$ My idea was to do it as ...
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28 views

Conditional expectation versus correlation

Consider two random variables $X$ and $Z$. Suppose $E(X)=3$ and $E(X|Z=z)=0$ for some realisation $z$ of $Z$. Does this imply that $X$ and $Z$ are correlated? Does this imply that $X$ and $Z$ cannot ...
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Alternating Conditional Expectations: Multiple regression transform

Alternating Conditional Expectations (ACE) is a non-parametric algorithm for multiple regression transform selection. It finds a set of transformed response variables that maximizes $R^2$ using ...
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Expected value of X given Y is less than some constant [duplicate]

Here is the problem I'm trying to work out: Let $v_b,v_s$ be jointly normally distributed random variables with pdf $f(v_b,v_s)$. I want to work out $E[v_b|v_s\leq\pi]$ for some constant $\pi$. Here ...
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Expectation of an Expectation

I need to solve two exercises: Calculate V[ui|xi] using E[yi|xi] and ui = yi - E[yi]. Calculate E[$y^2$|xi]. Information given for the exercise: E[$u^2$|xi] = V[yi|xi] E[$u^2$|xi] = V[yi|xi] ...
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conditional expectation with updated information

Let $$\epsilon_{t+1} = \rho\epsilon_t + \eta_{t+1}$$ $$E_t[r_{t+k}|\eta_t] = \phi^k \eta_t$$ Can we say that $$E_t[r_{t+1}] = \sum_{k=0}^\infty \phi^k \eta_{t-k}$$ Are there any conditions for this ...
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Deriving reward functions in Sutton & Barto

Does anyone know how the equations have been derived, I'm still learning probablity theory and expectations
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Truncated expectation of sum of independent random variables

Take three random variables $X$, $Y$, $Z$ s.t. $E[X]>0$, $E[Y|X]=0$, $Z = X+Y$. What can I say about $E[x| x> k]$ vs. $E[z| z>k]$ where $k>0$? Intuitively, the latter should be bigger ...
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145 views

How to generate data that have given conditional mean and conditional quantile using R?

Suppose I want to generate independent data $(y_{i},x_{i})$ such that the conditional mean of $y_{i}$ given $x_{i}$ is a quadratic function in $x_{i}$ and the $.25$ conditional quantile of $y_{i}$ ...
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Conditional expectation of functions of two random variables with inequality conditions

Consider general case first. Let $X$ and $Y$ be independent continuous random variables with known pdfs. What is expectation of $Z = \begin{cases} g_1(X, Y), & Y \geq X,\\ g_2(X, Y), & Y < ...
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35 views

Where is the error?

I am trying to compute expectation of $X\mathbb I_{[X+Y\le a]}$ where $a$ is a fixed positive integer, $X$ is discrete uniform random variable taking values from $1$ to $a$, and $Y$ another random ...
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3answers
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A generalization of the Law of Iterated Expectations

I recently came across this identity: $$E \left[ E \left(Y|X,Z \right) |X \right] =E \left[Y | X \right]$$ I am of course familiar with the simpler version of that rule, namely that $E \left[ E \...
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1answer
154 views

Conditional expectation function and the conditional indepence of residuals

im reading mostly harmless econometrics and im struggling with their brief notation and proofs. It seems to me like there is an inconsistency: First, they proove that if you split the dependent ...
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31 views

Correlation and expected values

Consider two random variables, $x$ and $y$. Denote the correlation between them by $\rho$. Assume that $E[x]$ is also a function of some parameter $\pi$ and is increasing in $\pi$. So if we increase $\...

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