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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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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2answers
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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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2answers
133 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 ...
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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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1answer
31 views
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 ...
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2answers
47 views
Conditional Probability and Expectation for Poisson Process
To solve part (a) I have $P(X_2 = k\mid X_1 = 1)= \dfrac{P(X_2 = k \cap X_1 = 1)}{P(X_1 = 1)} = \dfrac{e^{-2}}{e^{-1}}=e^{-1}$. Then for part (b), for simplicity, I let $X_2=X$ and $X_1=Y$, then $$E(...
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1answer
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Why is the conditional expectation the best predictor but only if we have the joint distribution?
If we want to predict one variable $Y$ based on another $X$, the best predictor is apparently $\mathbb{E}[Y \mid X = x]$. However, this apparently assumes two things:
The distribution is symmetric.
...
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Conditional probability of Negative Binomial R.V. given the SUM of its values
Suppose $\{z_{ij}\}$ are independent Negative Binomial random variables with means $\{\mu_{ij}\}$, with $i=1\dots I$ and $j=1\dots J$.
How do you find the (expectation of) conditional probability ...
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Applying boundary conditions and constraints to Gaussian process regression
When using Gaussian process regression (GPR) to predict $y$ over a domain, $x$, are there method(s) to impose particular conditions on the predictions? For example, if I know the prediction $y$ must ...
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1answer
22 views
Expectation conditional on self and others
I would simply like to know if:
$E[x_1|x_1,x_2]=E[x_1|x_2]$
or
$E[x_1|x_1,x_2]=E[x_1|x_1]=x_1$
or something completely different and why.
This is not homework. It came up because I'm trying to ...
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1answer
72 views
Expectation of standard exponential squared given sum of two standard exponentials
So I have been working on this question for a while and made some progress , but I run into a problem about the normalizing constant. The question is, for $X$ and $Y$ i.i.d. standard exponential, find ...
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1answer
73 views
Why does $\operatorname E(\varepsilon\mid x) = 0 \implies \operatorname{cov}(\varepsilon,x) = 0$?
I understand the intuition behind the question but I'm trying to prove it to myself with math.
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Expected value as an orthogonal projection
I'm reading a paper in which the expected value of a random variable, $\mathbb{E}[X]$, is characterized as an orthogonal projection. This is on page 10. I've seen the geometric interpretation of ...
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1answer
48 views
Conditional expectation of a vector
Suppose we have two random vectors $X=(X_1,X_2)^T$ and $Y=(Y_1,\dots,Y_n)^T$. I wish to find a simple definition or formula for
$$
E_{X|Y=y}[X]
$$
Intuitively, I think the following is correct:
$$
...
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1answer
26 views
Analytical solution to the covariance between a continuous and a categorical variable
Let $X$ be a continuous variable with mean $\mu$ and $Y$ be a categorical variable with event probability vector $\mathbf{p}$. I am trying to calculate $\operatorname{Cov}(X, Y)$.
I have the solution ...
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1answer
81 views
Multivariate Normal : expectation of X given Y is doubly-truncated
Let $(X, Y)$ be distributed as a multivariate normal with parameters
$$
\mu =
\begin{bmatrix}
\mu_X \\ \mu_Y
\end{bmatrix}
\qquad
\Sigma =
\begin{bmatrix}
\sigma_X^2 & \sigma_{XY} \\ \sigma_{XY} &...
1
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1answer
48 views
Independence and conditional distribution
In a problem that I'm solving I find that:
"Let data (yi,xi) be sampled randomly from a two-dimensional distribution such that y|x is N(ɑ,x^2σ^2)".
Are y and x i.i.d? maybe just identically ...
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1answer
26 views
Difference between averaging and ignoring the partial dependencies?
This question sparks from model interpretation/visualization. To graph the dependency of a function with >2 arguments, one often needs to ignore or average out some arguments.
Problem set
Hastie, ...
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1answer
43 views
Proof of contemporaneous exogeneity, and its implications for an AR(1) model
It can be shown by contradiction that exogeneity fails to hold for an AR(1) model.
Is there any proof that contemporaneous exogeneity does not fail to hold?
All I've come across is assuming it does ...
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0answers
35 views
Conditional mean and co-variance in $VAR(p)$ conditional on one lag only
Suppose I have a $p$'th order vector auto regression
$$\vec Z_t = F_1\vec Z_{t-1}+F_2\vec Z_{t-2} + \cdots +F_p \vec Z_{t - p}
+ \vec \epsilon_t,\qquad \vec\epsilon_t\sim N_q(\vec0,Q)$$
where $...
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0answers
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Understand conditional expectation w.r.t. sigma -algebra [duplicate]
When a random variable $X$ is discrete, the definition of conditional expectation of $X$ with respect to a decomposition $\mathscr D$ is
$$
E[X|\mathscr D] = \sum_{i = 1}^m x_i \sum_{j = 1}^n P(X|\...
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1answer
50 views
How to deal with expected value in the context of time series?
For example, in this MA(2) model,
$y_t = u_t + \phi u_{t-2}$
$u_t$ is identically, independently, normally distributed with a mean of 0 and a variance of $\sigma^2$. (Does variance matter here?)
I ...
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0answers
35 views
Questions relating the definition of conditional expectation $E[g(X)|M]$ where is $X$ is random variable and M is an event
I saw the following definition of conditional expectation from a book:
if M is event and X is continuous random variable then we define:
$$E[X|M]=\int_{-\infty}^\infty xf(x|M)dx$$
Which is the ...
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1answer
118 views
Estimate E(Y|X,Z) from E(Y|X) and E[Y|Z]
Can I estimate $E[Y|X, Z]$ if I know $E[Y|X]$ and $E[Y|Z]$?
As an example, let's say I have a model where
$X\sim N(0, 1)$,
$Y = aX+\epsilon_Y, \epsilon_Y \sim N(0, 1)$,
$Z = bY+\epsilon_z, \...
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0answers
15 views
Calculate target variance contionaly to continuous variable
I'm trying to get the conditional variance of my target Y conditional to my inputs X. To do this I first discretized my inputs variable into 1 qualitativ variable and then I computed the variance of Y ...
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2answers
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Generate Data where outcome is conditional on independent variables [closed]
I want to generate a synthetic dataset {Y, X1, X2}. Independent random variables X1 and X2 follow bernouli distribution where probabilities for X1 and X2 are known.
Whereas, outcome variable Y needs ...
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1answer
27 views
Prove convergence in distribution, probability, or quadratic mean for a sequence of binary variables that depend on another binary variable
Suppose that $X$ has the support set $\{1, -1\}$, and $P(X = 1) = P(X = -1) = 0.5$.
Suppose that $X_n$ has the support set $\{X, e^n\}$, and
$P(X_n = X) = 1 - \frac{1}{n}$
$P(X_n = e^n) = \frac{1}...
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0answers
58 views
Proving covariance equals zero given a specific conditional expectation
I'm trying to prove the following:
Given $𝐸[𝑋|𝑌 = 𝛽] = 𝐸[𝑋]$ for any value of $\beta$, prove that $\operatorname{Cov}(𝑋,𝑌) = 0$;
So I was thinking to start with the definition of $\...
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0answers
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Expected value of quotient of Poisson distributions
Let $X$ and $Y$ be independent random variables such that
$X \sim \text{Poisson}(\lambda \cdot c)$ and $Y \sim \text{Poisson}(\lambda \cdot (1-c))$, where $c$ is a real number in $[0, 1]$.
Is there ...
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1answer
56 views
Why can you not bypass the strong ignorability/unconfoundness assumption via iterated expectations?
Suppose we have that $\left(Y(1), Y(0)\right)$ are potential outcomes with $X$ being the covariate and $Z$ the treatment assignment. Typically in causal inference, one will assume strong ignorability ...
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1answer
55 views
Show $E[ (Y - E(Y|X)) (E(Y|X) - h(X))] = 0$
Show that $E[ (Y - E(Y|X)) (E(Y|X) - h(X))] = 0,$ where $X, Y$ are random variables with constant means and $h(x)$ is an arbitrary function.
So far, I have expanded out the expectation and used ...
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3answers
190 views
If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$
Question
If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$.
Attempt: Please check if the below is correct.
Let ...
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1answer
146 views
Conditional Expectation of pdf
Wish to identify what I'm doing wrong when finding the $\operatorname E(X\mid Y=5)$ of the following:
$$f(x, y)=\begin{cases}
1/6 & \text{if } 0<x<2, 0<y<6-3x \\ 0 & \text{...
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0answers
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Clarifying a proof of a particular paper on Steins Estimator
I am trying proving result (5.4) of the following paper. Its a paper on Steins estimator on spherically symmetric cases.
The doubt is a s follows:
Given
$$X|\theta\sim \mathcal{N}(\theta,I)$$ ...
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0answers
45 views
Linear Prediction and Linearity of CEF
I am revisiting the basic notions of linear regression and stumbled upon the following idea in Cameron and Trivedi's Microeconometrics book:
However, for the conditional mean to be linear in x, so ...
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1answer
32 views
Expectation of residuals in Zero Intercept Model
We know that the summation of residuals in a regression through origin model is not necessarily 0. Does that imply that Expectation of Residuals is not necessarily 0? CLRM still holds, so should that ...
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1answer
39 views
What does an expectation with respect to a policy mean in the reinforcement learning value function
I would like to know what the formal definition of the following expression is
$$
V_\pi(s) = \mathbb{E}_{\pi}(G_{t+1} | S_t =s)
$$
What does it mean to have the policy in the subscript? How would I ...
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1answer
93 views
Is the parameter vector of an indentifiable distribution of a transformed random vector always a subvector…?
I would like, after further considerations about this problem, to reformulate this question of mine again. I kept a record of the past words and remarks as the appendix below. I think the question ...
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1answer
114 views
How to calculate the integral of Normal CDF and Normal PDF?
I'm trying to find $\int_{\frac{a-b}{B}}^\infty\Phi\left(tA+ABx\right)\phi(x)\,dx$ where
$A = \frac{\sqrt{\gamma_{3}+\sigma_3^2}}{\gamma_{3}},\ B = \frac{\gamma_{2}}{\sqrt{\gamma_{2}+\sigma_{2}^...
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0answers
38 views
When to use all 3 variables in a graph to estimate a conditional expectation of 2
The title might not be perfect. But here goes:
Suppose there are 3 variables $(A,X,Y)$. And they have the following dependencies :
$\Pr(Y,A,X)=\Pr(Y\mid A,X)\Pr(X\mid A)\Pr(A)$
$A \rightarrow (X,Y)$...
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1answer
56 views
Conditional expectation of Poisson process given number of events
Let $\{N(t), t\geq 0\}$ be a Poisson process with rate $\lambda$, $S_n$ the instant of the $n$-th arrival and $T_n$ the $n$-th interarrival time, that is, $T_n = S_n - S_{n-1}$, $n \geq 1$.
Now ...
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1answer
28 views
Simple Appplication of Law of Iterated Expectation
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 ...
4
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1answer
40 views
Integral from the Adversarial Spheres paper (maximum of the difference between a constant and a normal random variable)
I'm trying to follow a proof in the Adversarial Spheres preprint on arXiv. The proof requires the computation of the integral in Appendix F, page 14:
$$\mathbf{E}\left[\max\left(\sqrt{2}\left(\frac{\...
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2answers
175 views
Conditional probability for consecutive Bernoulli trials
Independent trials, each of which is a success with probability $p$, are performed until there are $k$ consecutive successes. Let $N_k$ denote the number of necessary trials to obtain $k$ consecutive
...
3
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1answer
55 views
Expected value conditional on a function
Let $X$ and $Y$ be random variables. What is the relationship (if any) between $E[Y|X]$ and $E[Y|g(X)]$?
I have been trying to Google or look in books but I'm having trouble even articulating this ...
4
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2answers
74 views
One-sided measurement error: $\widetilde{X} = X - \eta$, $\eta\geq0$. Is $E[X|\widetilde{X}=\widetilde{x}]$ increasing in $\widetilde{x}$?
Let $X\geq0$, $\eta\geq0$ and $X,\eta$ independent. We measure $X$ with a one-sided error: $\widetilde{X} = X - \eta$.
Is $E[X|\widetilde{X}=\widetilde{x}]$ increasing in $\widetilde{x}$?
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1answer
44 views
The expectation of the $K^{th}$ raw moment of $X$ given $X$: $E(X^k|X)$
Intuitively I believe $E(X^k|X)=X^k$, $k$ is a non-negative integer. One obvious special case is when $k=1$. Anyone has an idea how to prove it?
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1answer
30 views
Do discriminative models model conditional expectation?
In Machine Learning classic models like MLP, Logistic Regression or Linear Regression are called discriminative models.
I frequently read that those models estimate the conditional probability ...
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0answers
85 views
Taylor Series Expansion of Unconditional Expectation
We know that the best 1st order approximation of an unconditional expectation is the following-
$$E(y|x)=(E(y)-\beta E(x))+\beta x$$
where $\beta=\frac{\operatorname{Cov}(y,x)}{\operatorname{Var}(x)}...
