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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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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48 views

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 ...
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Conditional expected value for dependent Log-normal Distribution

I'm trying to find the expected value of a FX derivative which is log normally distributed, dependent on another derivative, but do not have the direct co relation between them. Instead I'm given the ...
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iterated expectation conditional on two variables

How to prove that $E[Y]=E[E[E[Y|X_1, X_2]]]$ ? PS. I don't see how $E[E(Y|X_{1},X_{2})|X_{1}]=Y[Y|X_{1}]$ and $E[Y]=E[E(Y|X_{1})]$ can be used here. But it feels close. Please help, I'm stuck PPS. ...
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Is E(Y|X) a function of Y?

I'm slightly confused by a question I have encountered, it's a true or false question stating: 'The expression $E_{(Y|X)} (Y|X)$ is a function of $Y$.' My automatic instinct was, well yes of course ...
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Apprehension in a solved problem from Feller

This one is a solved example from Feller. Spores of the Fungus are produced in chains of eight.The chain may break into several parts into projectiles containing 1 to 8 pores. Find the ...
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Expected number of trials - Bayesian

ref: 2.11 Question 2. Gelman et al.: Bayesian Data Analysis 2nd ed Consider two coins, $C_{1} \text{ and } C_{2}$ with the following characteristics: $Pr(heads|C_{1}) = 0.6$ and $Pr(heads|C_{2})$ = ...
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Loss Size Index Function of A Lognormal Random Variable

I have this tutorial question and I've gone through the solutions, getting all but one line of working. I broke down the question to this point but I can't seem to get out the following. So Loss Size ...
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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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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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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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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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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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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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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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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 random variables that could be discrete or continuos and ...
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109 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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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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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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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 $c$, ...