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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Does the expectation of a quantile equal to the quantile of expectations?

Let $X$ be a random variable with finite expectation $E(X)$, and let's denote $X_{90}$ the 90% quantile of its distribution, meaning: $$P(X<X_{90})=0.9$$ Now, let Y be another random variable and ...
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Comparative statics for conditional expectations

Let $f\left(x,y\right)\in\left[0,\frac{1}{2}\right)$ a function such that $\frac{\partial f}{\partial x}>0$, $\frac{\partial f}{\partial y}<0$, and $\frac{\partial^2 f}{\partial x \partial y}<...
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What is the cubic expectation (third-order moment) of a complex gaussian vector (say, E[$aa^{T}a$])?

Note: I also posted this question on MATHEMATICS. For a real gaussian vector, an explicit formula for the cubic expectation can be found in Matrix Reference Manual (search 'Cubic Expectations' in this ...
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In a time series $x_t, x_{t-1},...,$, why is $E[x_t|x_t, x_{t-1},...]= x_t$?

Sorry if obvious but in a time series $x_t, x_{t-1},...,$, why is $E[x_t|x_t, x_{t-1},...]= x_t$? I don't really get what the random variable $x_t|x_t, x_{t-1},...$ represents? What I find ...
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Notation of expectation with conditional in subscript

Inside the book "The Elements of statistical learning", I stumbled upon the following notation (Ex. 2.7) $$E_{\mathcal{Y|X}}(f(x_0) - \hat{f}(x_0))^2$$ where $\mathcal{X, Y}$ are two random ...
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Simplify E[E[Y|X]|Z=z] in terms of E[Y|Z=z]

Let $Y, X, Z$ be random variables. It holds that $E[E[Y|X,Z]|Z] = E[Y|Z]$. See e.g. here Theorem 1 viii But I am particularly interested in the expression $E[E[Y|X]|Z]$. Is there a way to relate it to ...
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What's wrong with this conditional probability working?

A past exam paper for my course (BSc Mathematics, second-year module in statistical inference and modelling, unpublished) has a question, Let $(X,Y)^T$ be a bivariate random variable with joint ...
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Conditional expectation calculation with a binary variable

Suppose $X=\{0, 1\}$ is a binary random variable, $X$ and $Y$ are independent, $Z$ is another random variable. I get \begin{equation} E\left(\frac{X}{E(X)}Z|Y, X\right)=\frac{X}{E(X)}E(Z|Y, X). \qquad\...
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Expected Prediction Error for 0-1 Loss Function

In ESL on pages 20 and 21, we have a derivation of expected prediction error of a classification rule $\hat{G}(X)$: $$ EPE(\hat{G}) = E_X\sum_{k=1}^{K}L[\mathcal{G}_k, \hat{G}(X)]P(\mathcal{G}_k|X) $$ ...
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Exponential random variable X with a uniform random variable as its parameter

$$X\ \sim Exp(U) ~ and\ U\ \sim U(0,1) $$ The question asked for the value of $ P(X\geqslant 1)$ I saw the solution and it went like this: $$P(X\geqslant 1) = E[P(X\geqslant 1)|U] = E[e^{-u}] = \int_{...
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Loss Size Index Function of A Gamma Random Variable

I'm trying to prove that the loss size index function of a Random Variable Y, which is distributed as a Gamma Random Variable ($Y \sim Γ(γ,c)$) has the following expression: $$ I(y) = \frac{\textit{G}(...
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Showing that $E[\hat{\tau}_D] = P(n_D > 0)\tau_D$ and $\vert E[\hat{\tau}_D] - \tau_D\vert \leq \tau_D(1-\frac{N_D}{N})^n$

Consider the following double sampling scheme: We have a population of size $N$ with variable of interest $y_i$ for each $i \in \{1,\dots,N\}$, and (fixed) subpopulation $D$ of size $N_D$. Let $S$ ...
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Linear Regression : why expected value of response

I am really confused... why linear regression is modelling the expected value of response(or conditional expected value)? If we don't use mean square error as the loss function to minimise, is it ...
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Condition expectation of $X_1|\bar{X}$ [duplicate]

I've just learnt about conditional expectations and I'm confused about how we evaluate $E[X_1|\bar{X}]$ where $X_i\sim N(\theta,1),1\le i\le n$ and hence, $\bar{X}\sim N(\theta,1/n)$. Can someone ...
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Conditional expectation conditioned on an Indicator variable

Suppose I have a random variable $u$ that is standard uniformly distributed. And I have an indicator variable $S_{i}=1\left(V_{i}>0.5\right)$. Now I am interested in the following conditional ...
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Is there a function $g(X,Y)$ such that $E[g(X,Y)|X,Z]$ is constant in $X$?

More precisely, consider three random variables $X,Y,Z$, with $f(Y|X,Z)$ being the density of $Y$ conditional on $X,Z$. Say that this density is "regular" enough (e.g., continuous in all ...
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Identification of Richer Conditional Expectation from Less Rich Ones

Let $X_1,X_2$ be uniformly distributed and suppose that we know the quantities $E[Y | c_1 X_1 + c_2 X_2 = v]$ for all values of $c_1,c_2,v$ (i.e. we know all of the expectation of $Y$ conditional on ...
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Integral of conditional expectation over an event B

Could someone help me understand this equality? Let $\xi$ be a random variable. $\int_B(\frac{1}{P(B)}\int_B\xi dP)dP=\int_B \xi dP$ for any event $B$. How do we go from the integral over an event $B$ ...
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How to apply the conditional expectation of the multivariate normal distribution to fill gaps in data?

I have a data matrix $X \in \mathbb{R}^{m \times 4}$, where $m$ is any number of rows, whose data follow a multivariate normal (MVN) distribution. Suppose that for a given row $i$, the data for the ...
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Issue with Casella&Berger derivation of EM likelihood equality

In the explanation of the EM (Expectation maximization) algorithm p.328 in the book "Statistical inference" by G. Casella and R. Berger, 2nd edition, they present the following: $\mathbf{Y} =...
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Suppose $X, Y, Z$ are random variables. By Tower rule, $E(X) = E(E(X|Y))$. Is $E(X) = E(E(X|Y, Z))$?

Suppose $X, Y, Z$ are random variables. By Tower rule (iterated expectations), $$E(X) = E(E(X|Y))$$ My question is, is $$E(X) = E(E(X|Y, Z))?$$ My attempt (assuming $X, Y, Z$ are discrete r.v.): \...
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A problem in convergence and limit

For any non-negative integer $n$ and some finite $r$, we introduce the notation $n_k$ which indicates the number of $\{X_1, X_2, \cdots , X_n\}$ belonging to the $k$-th distribution type, for $k=1, 2, ...
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Finding $E[\min(X^2, 1)]$ with $X$ being standard normal

I'm trying to calculate the value of $E[\min(X^2, 1)]$ with $X \sim N(0,1)$. My attempt is that \begin{align} \begin{split} E[\min(X^2, 1)] &= E[1 | X^2 \geq 1] \cdot P(X^2 \geq 1) + E[X^2 | X^2 &...
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"If true propensity score is constant, then the superpopulation covariate distributions are identical in two treatment groups"?

This is a statement made in Rubin's Causal Inference Sec 12.5.1. I would take it under assumption of unconfoundedness framework. "If the superpopulation covariate distributions are identical in ...
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conditional expectation of random vectors

For a random vector $X=(X_1, ..., X_n)^\intercal$ the expectation value can be written as $\mathbb{E}[X] = (\mathbb{E}[X_1], ..., \mathbb{E}[X_n])^\intercal$ according to equation 2 in https://en....
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A new convergence problem for the conditional expectation

You have risks $X_1$, $X_2$, ... (they are assumed to be independent, but not necessarily identically distributed) and $S_n= X_1 + X_2 + \cdots +X_n$ QUESTION: under what reasonable conditions do we ...
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Conditional survival function in landmark analysis

In H.Putter & H.C. van Houwelingen's paper "Understanding Landmarking and Its Relation with Time-Dependent Cox Regression" the authors state that the conditional survival function, given ...
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What is the relationship between the conditional expected value, OLS and projection in linear regression?

I am learning about the linear regression. I was taught it from two perspectives. The first one was about an equation connecting the conditional expected value and the predictor. I saw the nice graphs ...
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How to formally define a conditional distribution conditioning on an event of probability zero?

Given $[X,Y]\sim N(0,I_2)$, a intuitive guess of the value of $P(X=x|\{X,Y\}=\{x,y\})$, where $\{\}$ means unordered set, is $1/2$ by symmetry. This type of notations is typically applied in ...
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Conditional expectation of two correlated RVs

$X$ and $Y$ are two correlated random variables. I am trying to estimate $E(X\mid Y)$ given $E(X)$, $E(Y)$, $\rho(X,Y)$, $\sigma(X)$ and $\sigma(Y)$. Could someone point me how to go about it. What if ...
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Show that two white noise definitions are equivalent

Given $(\varepsilon_t)\sim WN(0,\sigma^2)$ a white noise. By definition $$E(\varepsilon_t)=0,\,\, E(\varepsilon_t^2)=\sigma^2 \quad \forall t$$ and $$E(\varepsilon_t \varepsilon_s) = 0, \quad s\neq t$$...
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A question about conditional expectation involving independence

If the vector $(u,v)$ is independent of the vector $x$, then I would like to show that $$E(u|x,v)= E(u|v)$$ The only thing I can derive from the definitions is that if $(u,v)$ is independent of $x$, ...
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Consequences of Strict Exogeneity in a i.i.d. random sample

In the context of Strict Exogeneity on the classical regression model $$y_i = \beta x_i+ \varepsilon_j$$ we have: $$E(\varepsilon_i | x_i , x_j )=0,\quad \forall i,j$$ Under the linearity assumption, ...
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Conditional expectation of normal distributed variable with inequality condition [duplicate]

I am wondering how you compute the conditional expectation of a normal distributed variable X with mean and variance known. So more specifically: E(X|x<0) Is there a specific formula for this ...
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Conditional Gaussians in Infinite Dimensions

The law of the conditional Gaussian distribution (the mean and covariance) are frequently mentioned to extend to the separable Hilbert spaced valued case, i.e., for $(X,Y)$, $$ \mu_{X|Y=y} = \mu_X - ...
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Confidence Interval for Expectation using covariance matrix [duplicate]

Say I have a regression model as follows: $\ \hat{y}_i = \hat\beta_0 + \hat\beta_1x_1 + \hat\beta_2x_2, n = 79$ and I have the following covariance matrix $\ \begin{bmatrix} intercept & 6.5949972 &...
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Distribution of X given Y>0 with X and Y jointly normal [duplicate]

Suppose $X$ and $Y$ are jointly normal with mean $(\mu_X,\mu_Y)^T$ and covariance matrix $\begin{pmatrix} \sigma_x^2 & \rho \\ \rho & \sigma_Y^2\end{pmatrix}$. How can I compute the ...
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Relationship between two conditional expectation functions where one of them has a constant

I have two conditional expectation functions: E(log Y|X) = β0+β1X and E(log cY|X) = γ0+γ1X What is the relationship between β0 and γ0 and between β1 and γ1? Since there is a constant on Y, E(log cY|X) ...
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Inference on a selectively revealed sample

I think this question may be related to cryptography, so I may have the wrong stack exchange, but I am not really sure. Suppose there are two people Sam and Pam. Suppose we have a distribution, a set ...
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Suppose $Y \sim Ber(X)$ where $X = F(Z)$ and $Z \sim N(\mu, \sigma^2)$. What is the expected value $E[Y-X]$?

Suppose $Y \sim Ber(X)$ where $X = F(Z)$ and $Z \sim N(\mu, \sigma^2)$. $F$ denotes the cdf of some continuous random variable. What is the expected value $E[Y-X]$? \begin{align*} E[Y-X] &= E[Y]-...
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What is the theoretical justification for alternatives to MSE minimisation?

I'm trying to wrap my head around the connection between statistical regression and its probability theoretical justification. In many books on statistics/machine learning, one is introduced to the ...
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Calculate $E [ Z_{t-1}| X_{t-1}]$ in an ARMA process

Suppose I have an ARMA(1,1) model: $$X_t = \phi X_{t-1} + Z_t + \theta Z_{t-1}, \quad Z_t \sim WN(0,\sigma^2)$$ Indeed, I want calculate $E[X_t | X_{t-1}]$. For this: \begin{align} E[X_t | X_{t-1}] &...
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Conditional expectation of this stochastic process?

I'm just beginning to learn about stochastic processes and encountered this very elementary problem that confused me a bit: We toss a coin that lands on Head with probability $p$ and Tail with $q=1-p$....
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Is there a relationship between a regression's conditional mean of 0 and its 0 correlation with the error term?

In regression analysis, when we impose the exogeneity assumption, we express the assumption using the zero conditional mean condition. That is, $\mathbb{E}(u|x)=0$ is equivalent the statement "$x$...
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Rao-Blackwellisation using non-sufficient statistics

The following is given as a remark in chapter 7 of Introduction to mathematical statistics Hogg and Craig, 8th edition. (It is mentioned as "Remark 7.3.1") Now, I do understand that the ...
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Correct interpretation of the zero conditional mean assumption in the linear regression model

In many books the population linear regression model $Y = \langle \beta, X \rangle + U$ has the following "zero conditional mean assumption" (here $Y,X,U$ are random variables; let's also ...
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problem with filtrations for symmetric binomial stochastic process

$(a)$ Using symmetric binomial stochastic process up to $3$ period describe how information evolving over time gradually removes uncertainty over time. $(b)$ Describe how the process at $t=3$, that is,...
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Please clarify Bayesian calibration of the posterior mean

In the book Bayesian Data Analysis, the authors state on page 128: The concept of calibration of the posterior mean [is] the Bayesian analogue to the classical notion of bias. They define the ...
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Conditional expectation of X given X+Y [duplicate]

X and Y are two independent variables, X ~ exp(a), Y ~ exp(a). I need to find E(X|X+Y). I tried to calculate by definition, but it did not lead to success. Maybe there is another, more convenient way ...
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Expectation of a normal random variable when conditioning on a correlated normal random variable being above a threshold

Suppose $X$ and $Y$ are correlated with correlation coefficient $\rho$. They are jointly normal with means $\mu_X$ and $\mu_Y$ respectively. Then what is $E[X | Y \geq T]$? Feel free to add additional ...
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