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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When do we have that $E[X_{T+1} | X_T] = X_T$?

Given a filtered probability space $(\Omega, \mathscr{F}, \{\mathscr{F}_n\}_{n \in \mathbb{N}}, \mathbb{P})$: Let $X = \{X_n\}_{n \in \mathbb N}$ be a $(\{\mathscr{F}_n\}_{n \in \mathbb{N}}, ...
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Minimizing MMSE over positive random variables

Let X be a random variable with a finite second moment. We know that: Argmin E(X-Y)^2 = E(X|g), Where the minimum is taken over all g-measurable random variables Y. How can I find argmin E(X-Y)^2 ...
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42 views

Expectation of two identical lognormal distributions

I would like to compute the conditional expectation (on an interval from $c$ to $\infty$) of the minimum of two log normal distributions. Denote $X_1$, $X_2 \sim LN(0, \sigma)$, the associated ...
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expectation of conditional expectation

Given $(X,Y)$, 2-dimensional probability vector, and let $g: R^2 \rightarrow R, E[g(X,Y)^2 ] < \infty$ and $h:R \rightarrow R, E[h(X)^2] < \infty $, prove the following: ...
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Conditional expectation of random vector given one variable

I would like to find the conditional expectation $E(Y_i|X_i)$. Here $Y_i=\mu+\beta X_i+\gamma X^{T}+\epsilon_i$ where $X=(X_1,X_2,\ldots, X_n)$ is a $n\times 1$ vector, $X_i$ is iid random variable ...
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38 views

Example of Conditional Expectation in Markov Chain

My question is from the book Introduction to Probability Models, 10th edition, by Sheldon Ross. Here is a example in the book. Consider a Markov chain with states $0, 1,\cdots , n$ having $P_{0,1} = ...
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question about the conditional error term in linear regression

Suppose we are given $n$ i.i.d. random vectors $\{ y_i,X_i \}$ where $y_i$ is a random scalar and $X_i$ is a random vector. Further suppose that $\epsilon_i$ is a linear function of $\{ y_i,X_i \}$. ...
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Asymptotic conditional expectation

Problem Setup Let $\{X^d_1, X^d_2, \cdots, X^d_n\}$ be a $d-$dimensional zero-mean, i.i.d. random variables. Let $S_n^d$ be $$ S^d_n = \frac{\sum_{i=1}^n X_i^d}{\sqrt{n}} $$ Let $Y^d$ be a ...
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Conditional Expectation of a function and another Conditional Expectation

Suppose $\{X_1,X_2,Z\}$ is a vector of 3 real valued continuous random variables with compact support, $f_1(X_1,X_2)$, $f_2(X_1,X_2)$, and $g(X_1,X_2)$ are measurable functions with at least 2 ...
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Coordinate Ascent for Variational Inference: Deriving Updates

I am working with the following model and am attempting to derivate coordinate ascent updates using mean field variational inference: Sample $p_X \sim Beta(\alpha_1, \alpha_2)$ Sample $p_Y \sim ...
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38 views

Conditional expectation for non-gaussian variables

Let $A$, $B$ be two zero-mean random variables. Let the variance be $\sigma^2_A$, $\sigma^2_B$ and let the correlation be $\sigma_{AB}$. Consider the following expression :- $$ ...
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Deviation due to conditioning

Let $A$ and $B$ be random variables. Can we upper-bound the following expression? $$ \mathbb{E}\Big[\Big(\mathbb{E}[A|B] - \mathbb{E}[A]\Big)^2\Big] $$ The above looks classical research. However, I ...
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47 views

Zero conditional mean assumption (how can in not hold?)

Zero conditional mean of the error term is one of the key conditions for the regression coefficients to be unbiased. My question is: how can this assumption at all be violated if errors are equal to ...
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70 views

Estimating a function $f$ of a random vector $\mathbf{x}$ by a subset of the coordinates of $\mathbf{x}$ after a rotation of the input space

Suppose I have $$h=f(\mathbf{x})$$ with $f$ a deterministic function and $\mathbf{x}=(x_1,\ldots,x_n)$ a random vector of known distribution. I'm not using the capital letter notation for random ...
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Variance of the difference of two random variables compared to the difference of conditional expectation

Fix a probability space $(\Omega,\mathscr{F},\mathbb{P})$. Let $Y$ be a square integrable random variable $\mathbb{E}Y^2 < \infty$ and let $\mathscr{G}$ be a sub-$\sigma$-algebra of $\mathscr{F}$. ...
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49 views

Existence of the conditional tail mean

Does the existence of the first moment of a generalized Pareto distribution with support $[0,\infty)$ imply the existence (finiteness) of the conditional tail mean -- i.e. what in risk management is ...
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Finding conditional expected value

Given that X and Y are two independent exponentially distributed random variables with parameters a and b respectively. let Z = max(X,Y) find E[X|Z] attempt: I found that: P(Z=X) = b/(a+b) and ...
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Adding dependent random normal distributions and conditional expectations

My problem is as follows: Let $X, B_1, B_2, B_3$ be independent normal random variables with $\mu = 0$, $\sigma = 1$. Let: $Y_1 = X + B_1$ $Y_2 = 2X + B_2$ $Z = X + B_3$ Then, I had to find $Z' = ...
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Confused about “iterated expectations” step in a derivation

Let $X$ and $Y$ be jointly normal random variables with mean zero, variances $\sigma_x^2$ and $\sigma_y^2$, correlation $\rho$. Then, given a constant $\delta$: $\mathrm{E}\left[x\ |\ ...
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Approximating the conditional expectation in simulations

I am simulating stock returns, which are governed by the following equations $r_t = \mu + \delta r_{t-1} + \varepsilon_t$ $\sigma^2_t = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma^2_{t-1}$ ...
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Expectation from conditional expectations

Consider a real-valued random variable $X$ and a dichotomus random variable $Y\sim Be(p)$, all defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Could you help me to show ...
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Expectation of the distant future values conditional on the current information today

I am trying to understand a structural econometric model, in particular the one presented in "Stock J. H. and Wise,D. A., 1990. Pension, the option value of work, and retirement, Econometrica, 58 (5), ...
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I know $Y|Z \sim\mathbb{N}(Z,\sigma^2)$, but what is $Z|Y$?

Say I know the conditional distribution: $Y|Z \sim\mathbb{N}(Z,\sigma^2)$ Now, what if I reversed this though and wanted to find the conditional distribution $Z|Y$? From intuition I would expect ...
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Average conditional variance

Related to Explanatory power of variable. Given a data for 1d variable $Y$ and multidimensional variable $X$, what is the best way to compute average conditional variance of $Y$ given $X$: $$ \Bbb ...
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Expectation of a conditional density

I'm trying to figure out why the following equation holds: $$f_{Y}(y) = E(f_{Y|X}(y|X))$$ I have sort of "worked out" the RHS to be: \begin{align} f_{Y}(y) &= E(f_{Y|X}(y|X)) \\[5pt] ...
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Conditional Mean in Linear Regression

I have a question about linear regression in general. Suppose we have the following Data Generating Process:$$y_{i}=x_{i}\beta+\epsilon_{i}$$ Now, the thing is that from my understanding, each ...
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Deducing the regression function using the Squared Error Loss Function [duplicate]

I am reading Elements of Statistical Learning, and came across a deduction which I cannot understand. In the second chapter, the author defines the squared error loss and deduces the conditional ...
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Can I apply statistics to catch rare diseases or make decisions about fate?

Suppose I am 50 years old and a study found that in my city 10 people (CI=4 people) die at 50. Now it's July and I have a very malignant disseminated cancer and this year 14 people died at age 50 in ...
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Conditional Expectation of sum of uniform random variables?

Let $X,Y$ be independent uniform random variables on interval $[0,1]$. Can someone show me how to find the expectation of $X$ conditioned on $X+Y \ge (\text{say}) 1.3$? $$E[X | (X+Y) \ge 1.3]$$ ...
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X is stochastically increasing in Y $\implies$ $E\left[Y| X\right]$ increasing in X

I have two random variables, $X$ and $Y$. I know: $\text{pr}\left(X \le u | Y\right)$ is a decreasing function of $Y$ for all $u$. Does this imply that: $\mathbb{E}\left[Y | X\right]$ is increasing ...
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With given numbers $a_1, a_2,a_3,…,a_N$, let $W=\Sigma_{i \in s_n}a_i$. Calculate the mean and variance of $W$

From the set $R=\{1,2,3,...,N \}$, a set $s_n$ of $n$ numbers are chosen without replacement, $0<n<N$. With given numbers $a_1, a_2,a_3,...,a_N$, let $W=\Sigma_{i \in s_n}a_i$. Calculate the ...
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Consistent estimator of the expectation of a conditional probability

I'm stuck in a problem where I have distribution distribution $P(\boldsymbol{x})$, from which I know how to sample from (i.i.d.) and two functions of the random variable $\boldsymbol{x}$: ...
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Does the LHS of $E[X_n | \mathscr F_{n-1}]$ make sense even if $X_n$ is not integrable or adapted?

Let $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)$ be a filtered probability space. Then $X_n$ is a $(\{\mathscr F_n\}_{n \in \mathbb N}, \mathbb P)-$martingale if: $X_n$'s ...
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Expectation and Conditional Independence

This question is an aside from another question here on CV. We know that the expectation of the product of two independent random variables is the product of expectations, i.e., ...
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Estimating the conditional expectation

Consider the discrete random variables $G,W,N$ all defined in the probability space $(\Omega, \mathcal{F}, \mathbb{P})$, respectively with support $\mathcal{G}, \mathcal{W}, \mathcal{N}$. Consider ...
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Decomposition property of conditional expectation function

Let $Y$ be the outcome variable and $X$ and $U$ be covariates. The well known decomposition property of CEF is given by: \begin{align*} Y=E(Y|X,U)+e \end{align*} where \begin{align*} E(e|X,U)=0 ...
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Conditional expectation of normally distributed random variables

I am trying to figure out what $E[X_{i} {\large\mid} \sum_{i=1}^{N}{X_{i}^{2}}=c]$ is equal to. It is part of a proof in a book I am reading and the authors don't explain it more. They have that the ...
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Prove $E[|Z|] < \infty \to E[|Z_n|] < \infty$

Let $Z$ be an integrable random variable on filtered probability space $(\Omega , \mathscr F, (\mathscr {F_n})_{\{n \in \mathbb{N}\}}, \mathbb P)$ Define $Z_{n} := E[Z|\mathscr {F_n}]$. Show that ...
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Conditional covariance of AR(1)

I am trying to derive the conditional covariance of an AR(1) process, $\text{Cov}(y_t,y_{t+h})$. I have been trying to solve it taking in account the law of iterated expectations (LIE). However, I ...
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Expected Value in Poisson Point Process with Prior Knowledge

I have a setup with a homogeneous Poisson Point Process (PPP) of intensity $\lambda$ in $W \subseteq \mathbb{R}^d$ and a set $A \subseteq W$. I'm looking for the expected value of points in set $A$, ...
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Identifying Assumption D-I-D Conditional Expectation of Errors

I am reading this paper about Difference in Difference and I am confused by the statement regarding the conditional expectation of the error. I dont think I ever quite grasped what this value ...
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Poisson Processes: Conditional Expectation

For a Poisson process of an event with some rate r what is the probability to have an occurrence of this event in the next time interval given that n occurrences of this event happened in the k ...
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Conditional Variance Representation - Correct Form [closed]

Which of the below two representations of conditional variance is correct? Here, $X$ and $Y$ are random variables. ONE $$\text{Var}\left\{ Y|X\right\} =E\left[\left\{ Y|X\right\} ...
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Mean conditional on filtration

I'm quite new to time series analysis. I'm reading a text book, where the conditional expected value of a process $(X_t)$ is written as $E[X_t|\mathcal{F}_{t-1}]$, where $\mathcal{F}_{t-1}$ is called ...
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simple conditional expectation question

I came across the following in an article that I was reading. I cannot prove to myself that it is true or not? Any help would be greatly appreciated. Is is true that $E[x|y]=\rho y$, $E[x|z]=0$ ...
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Derivation of conditional expectation and variance of the AR(1) process

I have a question regarding the AR(1) process. I want to derive the conditional expectation $E(X_{(t)}| X_{(0)})$ and the variance $\operatorname{Var}(X_{(t)}|X_{(0)})$ of the AR(1) process: ...
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Does this type of diagram showing conditional mean have a name?

I'm trying to show the skill of a prediction $\hat{Y}$ relative to a true value $Y$ beyond simple statistics like MSE or $r^2$. I usually like looking at the scatter plot of $\hat{Y}$ vs $Y$ but in ...
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(1) A flight has a 1% chance of crashing. (2) It's scheduled to fly 200 flights that week. Does (2) add any info re: your risk?

And assuming that: a) This is that particular airplane's very first flight, ever (not just for the week). b) Any distributions involved should be discrete in the sense that they should be functions ...
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Is the equation “$Y=\mathbb{E}[Y|X] + error$” an identity?

Can I always use this equation to regress $Y$ on $X$, if I know the distribution of $Y$ to get an expression for the expectation term?
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Multivariate test for conditional (in)dependence

Suppose I have vector random variables $X,Y,Z$ of dimensions $n_x\times 1, n_y\times 1, n_z\times 1$ respectively. My goal is to test whether $X$ is conditionally independent of $Y$ given $Z$, given ...