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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$\log[E(y|x)]$ vs $E[\log(y)|x)]$

I have these two equations for elasticity: $$\frac {∂\log[\hat E(y|\mathbf X)]}{∂\log(\mathbf X_j)} $$ & $$\frac {∂[\hat E(\log(y)|\mathbf X)]}{∂\log(\mathbf X_j)} $$ I understand that if my ...
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53 views

Ordered Logit regression intrepretation

I hope the image is relatively clear... My dependent variable is the change in people's idea about whether it is a good idea, where it is the support they give to the same question minus the support ...
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Does the expectation maximization algorithm apply to this problem?

I have a sample of variables $x_i$, where each one is a function of known variables $y_i$ and $b_i$, and of an unknown variable $\alpha_i$. $$x_i=y_ib_i-y_i^{\alpha_i}$$ The density function of ...
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Conditional Expectation Bivariate Normal with OLS

I am working on the effects of omitting variables in a regression for my thesis. I already found a lot about the bias that is created by omitting variables, but not so much about what this does for ...
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26 views

Conditional expectation of error based on multivariate normal variables

I have the following situation; I know the "true" model behind my regression but I am intentionally omitting some variables/regressors to simplify the problem. Suppose the true model is: ...
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94 views

What's the maximum expectation of a conditional variance, $E[Var(X+Z_1 \mid X+Z_2)]$?

Let $X,Z_1,Z_2$ be 3 mutually independent RV's, with $Z_1, Z_2$ assuming $N(0,1)$ distribution. $X$ is constrained to have unit 2nd moment, i.e. $E[X^2] =1$, but may take arbitrary distribution. The ...
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56 views

Statistic to use to measure intersection probability in set of elements

Being two sets of elements: set a: ele1, ele2, ele3, ele4, ele5, ele6, ele7 set b: ele3, ele4, ele8, ele9 And is intersection: ...
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25 views

Sequential conditional simulation to avoid using a large covariance matrix

I would like to generate $S$ samples of a $T \cdot M$ dimensional vector, where $T$ is the number of time steps and $M$ the number of locations, i.e., the vector is a stack with $T$ values for ...
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32 views

Expectation of truncated normal X conditional on truncated normal Y

I am trying to derive: $E(X|a \leq Y \leq b)$ where $c \leq X \leq d $, $X$ and $Y$ are (doubly truncated) Gaussians with the same mean and different variance, and $a < c < d < b$ are the ...
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conditional/unconditional expectation and variance for an AR(1) process

We have an AR(1) process, $X_t=\alpha X_{t-1}+\varepsilon_t$ with $\varepsilon\sim(0,\sigma^2)$, $X_0=0$ and $|\phi|<1$. We have the conditional expected a value with respect to $X_{t-1}$: ...
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Truncated trivariate normal - conditional expectation

I am working on a paper in which I'd need to use the two following conditional expectations: $E(X_{1}|a \leq X_{2} \leq b)$ $E(X_{1}|a \leq X_{2} \leq b, a \leq X_{3} \leq b)$ where $X_{1}, X_{2}, ...
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133 views

Proof of Simplification of Conditional Expectation of Product of Random Variables

Could someone please provide detailed steps to prove or disprove the following? $E[XY\mid XY>k] = E[XE[Y\mid XY>k]]$ Here, $X,Y$ are independent random variables that could be discrete or ...
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25 views

Proof that Conditional Expectation of Sum is Sum of Conditional Expectations

\begin{eqnarray*} E\left[\left.\left(X+k\right)\right|\left(X+k\right)>0\right] & = & E\left[k\left|\left(X+k\right)>0\right.\right]+E\left[X\left|\left(X+k\right)>0\right.\right] ...
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Instrumental variable regression related question

In Lubik and Schorfheide (2007), they are saying that The monetary policy rule can be represented as $$ R_{t}=X_{t}'M\beta_{1} + Y_{2}'\beta_{2}+\epsilon_{t}^{R} \hspace{1cm}(1) $$ where ...
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35 views

Probabilities of conditional expectation values in uniform distribution

Let's consider a continuous random variable $X$ as follows: $f_X(x)=\left\{ \begin{array}{ll}\frac{1}{2}, &\mbox{if} \ x\in[0,1] \\ \frac{1}{4}, &\mbox{if}\ x\in(1,3]\end{array}\right.$ ...
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26 views

Double expected value, which comes first?

In the following equation, the outer expectation is over the distribution $X_i|T_i = 1$ $\tau|_{T = 1} = E(E(Y_i|X_i, T_i = 1) - E(Y_i|X_i, T_i = 0)|T_i =1)$ Are we taking the expected value of ...
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38 views

Conditional Expectation of a product and sum of three Gaussian Random Variables

I have a problem and I have no idea how to tackle it. Any help would be greatly appreciated. The problem is the following: Assume I have three mutually independent random variables $a$, $b$, and $c$, ...
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50 views

E[x|y] where y=cdf(x) and y is random

I have a grid approximation of a cdf, $F_x$. The cdf has support for $x>=0$ From there, calculating the $E[x]$ is straight forward with some std numerical integration techniques. In my case, ...
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104 views

Conditional expectation everywhere non-zero while unconditional one zero?

I have a real-valued random variable $X$ that takes on positive and negative values, and $$E(X)=0 \tag{A}$$ There is also another real-valued random variable $Y$, not independent from $X$, neither ...
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177 views

Order Statistics problem: why doesn't law of total expectation (Adam's law) work?

This is the problem The opening prices per share, $Y_1$ and $Y_2$, of two similar stocks are independent random variables, each with a density function given by $$f (y) = ...
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160 views

How can I maximize my chances on an exam?

I have an exam tomorrow, there are 18 topics, the professor gives only one. But it is not enitrely random how he gives the topic. First he gives an interval like 18-52, 30-62, 30-68 etc, then we ...
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47 views

How to compute $P(|X - E_Y[h(y)]| < c)$?

Consider a discrete random variable $Y$, a continuous random variable $X$, and a constant $c$. The goal is to find $$P(|X - E_Y[h(y)]| < c),$$ when we are only given $P(y)$, function $h(y)$, and ...
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Conditional moments of bivariate normal

Suppose that (X,Y) are bivariate normal with non-zero means and correlation. Is there any neat expression for $\mathbb{E}(X|Y>0)$?
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135 views

Is it true that $E[Y|W]=E[Y|E(X|W)]$, given that $W$ is $X$ measured with error

In Carroll's 2006 book "Measurement Error in Nonlinear Models", and on p38 it is stated without proof that: [One can] estimate the regression of $Y$ on $(Z, X)$ and then to substitute into this ...
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287 views

Conditional Expectation E[X] = E[X|Y<=a] + E[X|Y>a]

Is it generally true that for random variables $X$ and $Y$, regardless of being dependent or independent, that $E[X] = E[X \mid Y \le a] + E[X \mid Y>a]$ ?
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Regression Specification and Relation to Conditional Expectation Function

In their book, Mostly Harmless Econometrics, Angrist & Pischke introduce regression as an approximation to the conditional expectation function (CEF). They present (p 46) this equation: ...
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49 views

Minimizing absolute devitation using median

I need to prove that the expectation of absolute deviation is minimized by the median. We are given that $$med(y|x) = \beta_0 + \beta_1x$$ and $x$ can take on three values with positive probability: ...
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48 views

Conditional Expectation of Order Statistics

Given $X_1,...,X_n \sim f(x)$ How do I find $E(X_{(1)} | X_{(2)})$? Would I have to find the conditional pdf and integrate wrt x? I get the conditional distribution to be $f_{X|Y}(x|y) ...
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25 views

Finding the best predictor Brownian motion

I want to find the best predictor of $(B_3-B_2)(B_4-B_{\pi})$ given an observation of $B_1$ Where $B_t$ is brownian motion for time $t \geq 0$. I am not sure how to approach this. I know it will be ...
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choice of maximum likelihood over expectation maximisation

Given a probability distribution two common statistical measures are the expectation value and the maximum likelihood (equivalent to mean and mode?). My question is, given a probability distribution ...
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Expected number of trials

Consider independent trials, each of which is a success with probability p and derive the expected number of trials needed to obtain k consecutive successes by (a)conditioning on the time of the ...
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A step of a proof regarding the Nadaraya-Watson estimator

Let the data be $(y_i , X_i) $ where $y_i$ is real valued and $X_i$ is a q-vector. The regression function for $y_i$ on $X_i$ is $g(x) = E(y_i | X_i = x)$, we can write this as: $$y_i = g(X_i) + ...
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Conditional expectation in mixture distributions

I have a mixture distribution for observed lifetime data $(\delta_i,t_i,L_i)$, where $\delta_i$ is a censoring variable (1 indicating death, and 0 indicating censoring), $t_i$ is the observed lifetime ...
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Nadaraya-Watson Optimal Bandwidth

I am currently working on a statistical project where I need to estimate a conditional expectation $E[Y|X=x_i]$ using the Nadaraya-Watson estimator. For doing that, I have the sample ...
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Conditional Expectation for Forecasting Intervention Model

William and Wei: Time Series Analysis Univariate and Multivariate Methods, Second Edition, page 90, gives the conditional expectation for $Z_{n+l}$ ($l$ step forecast of $Z_n$): $$ \hat{Z}_n(l) = ...
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Comparison of two estimators based on mean squared error

I'm refreshing my knowledge of statistics, and I'm stuck on a problem that asks for a comparison between 2 estimators (unfortunately, I can't remember the source of the problem, but I do know its ...
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29 views

Conditional pmf and mean for geometric rv

I have a geometric r.v. p(1-p)^(n-1), n = 1, 2, 3, 4, ... I was able to figure out the conditional mean conditioned on X > a. My answer to this is E[X] + a. I am pretty sure that is correct but if ...
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165 views

An 'easy' exercise on conditional expectations and filtrations

I am struggling with the following exercise in the context of modeling information structure via filtration to evaluate contingent claims. I hope that someone can explain me how to derive the ...
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135 views

Marginal, joint, and conditional distributions of a multivariate normal

Let $Y$ ~ $MVN_3(\mu, \Sigma)$ where $\mu = (5,6,7)$ and $\Sigma = \begin{bmatrix}2 & 0 & 1\\0 & 3 & 2\\1&2&4\end{bmatrix}$ Find (a) The marginal distribution of $Y_1$ (b) The ...
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38 views

Cumulative distribution function of dependent variables

I have measured packet delays on several different radio channels $c_1, c_2, c_3$ and got three streams of delay data: $d_{11}, d_{12}, d_{13}, \dots$ $d_{21}, d_{22}, d_{23}, \dots$ $d_{31}, ...
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Simple question about finding conditional expectation

Let $X,Y \sim U[0,1]$ ($X,Y$ are independent), we want to find $E[X|X>Y].$ I tried a few approaches to the above problem, but am not confident in my answer. One approach is as follows. Note that ...
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Conditional expectation to define causal effect

I'm reading these notes which are discussing the NRCM approach to analyzing causal relationships, that is to say, treats the causal inference problem like a missing data problem (where the missing ...
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Conditional Expectation and Variance with Multiple Conditions

Question on the properties of conditional expectation - Is it true that $E [W(t) | W(s) - W(u)] = E[W(t) | W(s), W(u)]$ ? Context - To prove that for a Wiener process $W(t)$, $E[W(t) | W(s),W(u)] = ...
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Mulitvariate normal truncated conditional expectation

There is three-variable multivariate normal distribution. Denote 3 variables with $X_1$, $X_2$, $X_3$. Let $\mu_i$ be means, and $\sigma_i^2$ variances of respective variables, and let $\Sigma$ be ...
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Estimating conditional variance y|x

I am building a predictor for $y = f(x)$ using training samples ${(x_i, y_i)}$ (assume) drawn i.i.d from some distribution $p(x,y)$, by optimising the empirical L2-loss: $f(x) = argmin_f \; \sum_i ...
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How to compute the expectation of a normally distributed random variable given an imprecise signal?

Given $r\sim\mathcal{N}\left(\bar{r},\frac{1}{\alpha}\right)$ where $0<\bar{r}<1$ and an imprecise signal about $r$, $x_i=r+\epsilon_i$ where ...
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Conditioning on Two Variables

Does the following equality hold true? $E[Z|\{X,Z\}] = Z$ If not, then when will it hold true?
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Estimate E[x|A,B]: alternatives to bucketing for non-parametric estimation

I have a set of products. I would like to estimate Expected Value of items sold of the products wrt product price and age of the purchaser. One alternative is to assume a distribution and fit it. ...
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53 views

Can E[X|Y] be computed from E[X|Y,Z]?

Let X, Y and Z be 3 discrete random variables. Is the following true? $\sum_{i=1}^\infty P\{Y=i\} E[X|Y=i,Z] = E[X|Z]$
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125 views

Expectation on higher-order products of normal distributions

I have two normally distributed variables $X_1$ and $X_2$ with mean zero and covariance matrix $\Sigma$. I am interested in trying to calculate the value of $E[X_1^2 X_2^2]$ in terms of the entries of ...