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 expectation of two regions

In $\mathbb{R}^2$, define square region A with corners at ${(0.5,0),(1.5,0),(1.5,1),(0.5,1)}$ and and square region B with corners at ${(0,1), (1,1),(1,2), (0,2)}$. Suppose that random variables X and ...
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Projection operator: why squared norm of the sum of them is equal (or smaller) than the sum of the squared norms?

I am working through the proof of Lemma 2 in this paper (page 25, need it for my own research) and I am stuck at the very first step. Here, I will formulate a bit simplified version of this step. ...
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Conditional Expectation : How is E[E[xy|x]]=E[xE[y|x]]? [duplicate]

I read that "The law of total expectations states E[xy]=E[E[xy|x]]. By linearity of conditional expectations, E[E[xy|x]]=E[xE[y|x]]" but I am not able to understand the part "E[E[xy|x]]=...
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What is an example of a loss function that is not minimized by the conditional expectation?

From statistical decision theory we know that if we want to minimize EPE (Expected prediction error) it is sufficient to minimize the conditional expectation of the loss function. $f(x) = argmin_{c} ...
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Conditional expectations algebraic step

I am reading this article https://arxiv.org/pdf/1605.07723.pdf and I am confused about the step in the proof of lemma D.5 at page 19. I try to make the argument abstract, so that it should not be ...
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AR Unconditional vs Conditional Moments

I was hoping someone might be able to explain the intuition behind conditional and unconditional moments and specifically when I ought to us which. In the sense that: I first started with considering ...
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If conditional expectation is equal to unconditional expectation does that mean the random variables are independent?

If $E\left(Y|X\right) = E(Y)$ can we state that $X$ and $Y$ are independent? And vice verse, if $X$ and $Y$ are independent can we state that $E\left(Y|X\right) = E(Y)$?
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Estimate $E[X_1 | X_1>X_2>\cdots>X_k]$ with simulation

Suppose Random variables $(X_1,X_2,\cdots,X_k)$ are mutually independent, but not identically distributed. I want to estimate $E[X_1|X_1>X_2>\cdots>X_k]$ with simulation. I am wondering if ...
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Let X,Y be bivariate normal , what is E[X|Z] where Z = X + Y? [duplicate]

I am trying to understand how does expectation and variance looks when Let X,Y be bivariate normal I want understand E[X|Z] and Var[X|Z] when Z = X + Y
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Pulling out constants out conditional expectation

Given $p_t = p_{t-1} + \varepsilon_t$ and $\varepsilon ~ $i.i.d. with$~E(\varepsilon_t) = 0~\forall~t$ I wanna derive the conditional expectation for $p_t$ and I know it is $p_{t-1}$ since it is a ...
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Implications on the relation between signs of random variables

Consider a binary random variable $Z$ and a random variable $Y$. Suppose that the following relations hold $$ Z=1 \Rightarrow Y\in \mathbb{R}^{+}_0\\ Z=0 \Rightarrow Y\in \mathbb{R}^{-}_0 $$ In words, ...
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Show that a conditional expectation is zero

Consider two binary random variables $G,Z$ and a continuous random variable $\eta$. Assume that $$ \begin{aligned} & (A) \quad E(\eta|Z=1)=E(\eta|Z=0)=0\\ & (B) \quad Pr(G=1| Z=1, \eta)=1 \...
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Prove conditional expectation $E(A|X=x)$

We already know that $P(A|X=x)=\frac{P[A I \{X=x\}]}{P(X=x)}$, but how to prove $E(A|X=x)=\frac{E[A I \{X=x\}]}{P(X=x)}$? In other words, can I say $E(A|B)=\frac{E[A B]}{E(B)}$?
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Conditional mean and orthogonality

I am stuck with the proof that the conditional mean satisfies the orthogonality conditions. Say, we have Y as a scalar random variable with finite variance and X as random vector. Conditional mean of ...
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Expectation of sequence of Random Variables

$X_1, X_2, .....X_n$ is a colletion of Random Variables. They are said to be $\textit{multiplicative system}$ if, for any $1 \leq k \leq n$ and for any set of $k$ indices $1 \leq i_1 < i_2 ....i_k \...
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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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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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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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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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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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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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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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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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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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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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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 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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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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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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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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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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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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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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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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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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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 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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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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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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37 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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