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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Proof that when the conditional mean of Y does not depend on X, Y and X are uncorrelated

I've been struggling to understand the proof that when E(Y|X) = $µ_y$, then cov(Y,X) = corr(Y,X) = 0, as provided in Stock and Watson's Introduction to Econometrics, 4th Edition. The book says the ...
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Expectation of a random variable over disjoint intervals

I wanted to know if there is a formal way to show the following. Say I have a random variable X that takes value over the interval ...
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Show that the manhattan distance from the origin of an ubiased random walk in $\mathbb{Z}^2$ defines a martingale sequence

Consider the infinite lattice $N \times N$. A pebble starting at the origin walks at random, each time moving equiprobably to one of its four neighbors. Let $X_i$ be the distance from the origin, ...
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Conditional expectation of a normal variable given a lower bound [duplicate]

I saw the identity below but I'm not sure how to derive it. $$E[X\mid X>K] = \mu + \sigma \frac{\phi(z)}{\Phi(-z)} \text{ where } z = \frac{K-\mu} \sigma$$ I'm stuck at the following step: $$E[X\...
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Conditional expectation question

I'm looking at question 4 section 3 from this problem set, bottom of page 2. Repeated (more succinctly) here: There is a prize $V \sim \text{Unif}[0, 1]$ measured in millions of dollars. You can ...
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What is the general formula for conditional expectation and variance conditioned on 2 variables?

I am trying to solve the E[V|P,S] in Noisy Rational Expectation Equilibrium. We know that E[v|q]=E(v)+(cov(v,q)/var(q))(q-E(q)). I need to know the above formula for E[v|q,S] in addition to its ...
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How to estimate the nested conditional expectation?

Given the continuous random variables $X$, $Y$, $Z$, I want to estimate $\mathbb{E}\left[Y\mid X=x, Z\right]$ and $\mathbb{E}\left[\mathbb{E}\left[Y\mid X=x, Z\right]\mid Z\right]$. Are there any ways ...
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General and specific ways to compute the conditional expectation

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X: \Omega \rightarrow \mathcal{X}$ and $Y: \Omega \rightarrow \mathcal{Y}$ be random variables. If $X$ and $Y$ are absolutely ...
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Difference between $\mathbb{E}[Y|X]$ and $\mathbb{E}[Y|X=x]$

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X: \Omega \rightarrow \mathcal{X}$ and $Y: \Omega \rightarrow \mathcal{Y}$ be random variables. I have two questions comparing the ...
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Different formulations of the conditional expectation [duplicate]

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X: \Omega \rightarrow \mathcal{X}$ and $Y: \Omega \rightarrow \mathcal{Y}$ be random variables. I have a question comparing the ...
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When is the Conditional Expectation equal to the Best Linear Predictor for Binary Variables

Given random variables $X$, $Y$ and $Z$ where $X$ and $Z$ are binary, when is $E(Y|X)=BLP(Y|X)$? What about $E(Y|X,Z)=BLP(Y|X,Z)$? What about $E(Y|X,Z,XZ)$? This is what I have so far: $E(Y|X)= E(Y|X=...
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If $X$ and $Y$ are uncorrelated random variables, then under what condition is $E[X \mid Y] \approx E[X]?$

Suppose $X$ and $Y$ are real random variables that are uncorrelated. Now, uncorrelated does not imply independence, so $E[X \mid Y] \ne E[X]$. However, can they be said to be approximately equal? If ...
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Conditional expectation evaluated at function of random variable

Let $X, Y$ be continuous, real valued random variables, and let $f$ be a measurable function such that $f(X)$ is again a random variable. EDITED: How would the conditional expectation $\mathbb{E}[Y|f(...
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Homework question about Poisson Processes with conditional expectation

I think I have the first two questions figured out, but I'm stuck on conditional expectation. Consider a Poisson process, X = {X(t) : t ≥ 0} with rate λ = 2 per hour. 1) Covariance(X(5) - X(2), X(4) - ...
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MGF of the absolute Value of a Skellam RV

I am trying to derive the moment generating function for the absolute value of a Skellam random variable $Skellam(\lambda_1, \lambda_2)$ Suppose $X_1 \sim Pois(\lambda_1)$ and $X_2 \sim Pois(\lambda_2)...
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Variance of the product of two conditional independent variables

Now I know that the variance of the product of two independent variables $Y$ and $Z$ is:$\DeclareMathOperator{\Var}{Var}$ $\Var(YZ) = \Var(Y)\Var(Z) + \Var(Y)E(Z)^2+\Var(Z)E(Y)^2$ However I would like ...
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Find conditional expectation given symmetric r.v

How to find $E[sin(X)|cos(X)]$, if X - a symmetric random variable? My attempt: I tried to prove that $sin(X)$ is $\sigma(cos(X))$-measurable. Then it can be argued that $E[sin(X)|cos(X)]=sin(X)$. But ...
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What’s a textbook covering similar content to “Introduction to Probability Models” by Sheldon Ross?

I’m taking a class with a instructor using said textbook, and I find the explanations in it lacking. It’d be great if anyone can offer an alternative book covering similar content (i.e. conditioning ...
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Conditional expectation $E[V|U=u]$

Assuming have two variables $U \sim Uniform(0,1)$ and $V \sim Uniform(0,1)$, I want to calculate the conditional expectation $E[V|U=u]$ in R. Is there a specific ...
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Covariance between partitions of a normal distribution

A bit of a contrived example, but if I had a sample of $X_1,\dots,X_n \stackrel{iid}{\sim} N(\mu,\sigma^2)$ (in this case $\mu$ is unknown but $\sigma^2$ is known), and then calculated the arithmetic ...
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Jensen's inequality in EM derivation: expectation derivation

In Where does Jensen's Inequality come into the EM derivation? and in McLachlan & Krishnan (1997) - The EM Algorithm and Extensions, it was shown that \begin{equation} H(\theta|\theta^{(t)}) \...
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Multivariate normal, conditioned on inequalities

$X_1,\ldots,X_n$ are jointly normal random variables with some mean vector $\mu$ and covariance matrix $S$. How do I evaluate $P \left (X_1 \le x \mid \sum_{i=1}^nX_i \ge y\right )$? Is there a quick ...
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Derivation of expected value in REINFORCE policy gradient

This is a derivation in the book "Reinforcement Learning, an Introduction, 2ed" for the REINFORCE algorithm. By definition $q_\pi(s,a)=\mathbb{E}[G_t|S_t=s,A_t=a]$. I don't understand how ...
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Why are standard errors larger for conditional effects versus marginal effects?

I am estimating the conditional and marginal effects for a continuous by continuous interaction in a linear mixed effects model. The standard errors for the conditional effects are much larger than ...
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Simple Probability Question and Confusion

I was reading an MIT Problem Set and one of the first questions was confusing. It asked. If you flip a coin 3 times, determine the probability of events. Assume all sequences are equally likely. So I ...
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Expectation of next value conditioned on previous sample

I've been studying for my estimation class, and I can't wrap my head around the lecturers notes. The question is about empirical bayes credibility models, but my confusion boils down to this: Consider ...
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Is non-linear least squares the best approximation of the corresponding population conditional expectation in a least squared error sense?

Background Loosely adapting Angrist & Pischke Mostly Harmless Econometrics (A&P) section 3.1, suppose we have a conditional expectation function (CEF) $\mathbb{E}\left[Y_i|X_i\right]$ defined ...
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Why do we care more about test error than expected test error in Machine Learning?

In Section 7.2 of Hastie, Tibshirani, and Friedman (2013) The Elements of Statistic Learning, we have the target variable $Y$, and a prediction model $\hat{f}(X)$ that has been estimated from a ...
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How to invert/infer a parameter in nonlinear conditional expectation function

I wouldn't be surprised if this question has already been asked, as it sounds like a standard bookwork result. However, I'm not sure I know the language to describe it, and when I type in the the ...
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Expectation and Variance of Sum of dependent discrete variables

Q. Let X be a discrete random variable such that X = 0 with probability 0.5 and X = 1 with probability 0.5. Let Y be a discrete random variable such that Y = 1 when X = 1 and Y = 0 when X = 0. What is ...
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Computing the expectation of a product from the conditional expectation

Consider 3 random variables $Y, W,X$. Suppose that we know $$ (1) \quad \mathbb{E}(Y| X=x, W=w) \quad \text{ for each possible values $x,w$ taken by $X,W$} $$ Question: Can we compute from such ...
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Does conditional symmetry imply mean independence? [duplicate]

suppose I have two random variables $X$ and $Q$. $Q$ is conditionally symmetrically distributed about zero, i.e., its density satisfying satisfying $f(-q|X=x)=f(q|X=x)$ for every $q\in \Omega_{Q|X}$...
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Conditional Expectation Calculation

I want to calculate a conditional expectation $E[U|Y=y]$ where $U$ follows a normal distribution with mean $m$ and precision $h$ (variance $1/h$), and $Y=U+\epsilon$ where $\epsilon$ follows a normal ...
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265 views

Find Cov(x,y) as a function of x

Prove that in any bivariate distribution, $$Cov[x,y] = Cov_x[x,E[y|x]] = \int_x(x - E[x]) E[y|x] f_x(x) dx. $$ (Note that this is the covariance of x and function of x.) Source: Econometric Ananlyis (...
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Rao-Blackwellization in Black Box VI

In the paper, "Black Box Variational Inference," by Ranganath et al. (2013), the authors derive a Rao-Blackwellized estimator of the gradient of the evidence lower bound with respect to a ...
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161 views

Conditional expectation conditioned on function of a random variable

Let $X, Y$ be continuous, real valued random variables, and let $f$ be a measurable function such that $f(X)$ is again a random variable. How would the conditional expectation $\mathbb{E}[Y|f(X)=f(x)]$...
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Does independent marginal distribution indicate independent conditional expectation?

Let's say, $z=[z_1,...,z_d] \in R^d$ is a multivariate random variable, and $x\in R^d$ is another multivariate random variable, and $r(x)=[r_1(x),...,r_d(x)]=[\mathbb{E}[z_1|x],...,\mathbb{E}[z_d|x]] =...
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For a linear regression of Y on X, when is the regression of X on Y linear? When is it non-linear?

Suppose we have two random variables $X_1$ and $X_2$ that both have finite expectations, that may or may not be correlated, and that are known to be linearly related (i.e., I mean their sum forms a ...
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If $E(X|Y) = X$, what does this imply about the relationship between $X$ and $Y$?

Let $Z, K, V$ denote random variables, where $Z$ is binary, $K$ is categorical from 1-10, and $V$ is continuous. Let $X = P(Z = 1|V = v)$ and $Y_k = P(Z = 1|V = v, K = k)$. Now define $Y = (Y_1, \...
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Regular conditional distribution vs conditional distribution

What is the difference between the concept of the regular conditional distribution and the concept of the conditional distribution? Why do we need these two different concepts? Under which ...
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Lower bound on conditional entropy when conditioning on event

In Lemma 3.8 of this paper, the following is shown for jointly distributed random variables $Z$, $W$, where $Z$ takes values in $\{0,1\}^n$, and some event $\mathcal{E}$: $$ H[ Z \mid W ] \ge n - d\ \...
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Why isn't every nonparametric model with random model design an additive noise model?

Let $Y$ be a real random variable and $X$ be a real random vector. In a nonparametric model with additive noise, we assume the relationship $$Y = f(X) + \epsilon$$ for some unknown regression function ...
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Interpretation of coefficients in multivariate Poisson regression in terms of expectation

If I have a regression model $Y = \beta_0 + \beta_1 A + \beta_2 B$, where $A$ and $B$ are treatment indicators. Firstly, if I frame the definition of $\beta_1$ in terms of expectations, how do you ...
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Simplifying the conditional expectation $ E[X|Y]=(-a +Y-E[e|Y])/b$ to find the slope of the Reverse Regression line

I have a pretty basic question about conditional expectation that is stumping me. Consider the real-valued random variables $Y$, $X$ and $e$, where $E[e] = 0$ and $X$ and $e$ are independent. Assuming ...
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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]]?

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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139 views

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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