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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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Law of the unconscious statistician for conditional expectation and pushforward measure of conditional distribution

Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and $X:\Omega\rightarrow\mathcal{X}$ and $Z:\Omega \rightarrow \mathcal{Z}$ two random variables. The unconditional version of the law of ...
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Does the conditional expectation operator have an interpretable decomposition like the projection matrix does in linear algebra?

I'm trying to draw a parallel between the concept of projections in a finite linear space to an infinite linear space. Here is the set-up, first in the finite dimensional case, and then second in the ...
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GLMs and their conditional expectation and variance

Let the density of the distribution of response $y_i | x_i$ in GLMs denote as: $$f(y; \theta, \phi) = \exp\left(\frac{y\theta - b(\theta)}{\phi} + c(y; \phi)\right)$$ Then conditional expectation and ...
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What is the expectation of an arbitrary observation conditional on non-iid sample mean?

I'm specifically interested in the expectation of an arbitrary observation from a time indexed sample of correlated observations, conditional on the sample mean: $\mathbb{E}{[X_{t}\mid \frac{1}{T}\...
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Confusion about the notation $X\in\mathcal{F}_o$, where $X$ is a random variable and $\mathcal{F}_o$ is a sigma-algebra

In Durrett's Probability:Theory and Examples page 205 section 4.1, it has the following notation $X\in \mathcal{F}_o$ (see the picture below). I'm confused about this notation as $X$ is a random ...
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Applying the law of total variance

Say we have a sample of 100 normally distributed payments, with mean=1000 dollars and standard deviation= 100 dollars. 10% of these payments were made in error and should be refunded their full ...
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Expectation of the product of two random variables

I recently tried to derive a formula that I saw in a paper. The scenario was a follows: Let $X\in\lbrace 0,1\rbrace $ a.s. be a binary random variable and $Y$ be a continuous random variable. Let $a,b\...
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Conditional and unconditional mean in GARCH(1,1) model

Say I have a stationary time series and want to fit a GARCH(1,1) model. Does this mean that the conditional mean, which is used in GARCH, would always be the same as the unconditional mean of the ...
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Conditional expectation given the rank of the variable

Suppose that we have a random variable $X$ with distribution $F_X(x)$. Define the rank of the variable as $R = F_X(X)$. What can we say about $\mathbb{E}[X \mid R]$? If $F_X(X)$ is strictly ...
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How to come up with an example that $E(\epsilon|z,\eta)=E(\epsilon|\eta)$ and $E(\epsilon)=0$ do not imply $E(\epsilon|z)=0$?

I'm trying to come up with an example showing that $E(\epsilon|z,\eta)=E(\epsilon|\eta)$ and $E(\epsilon)=0$ do not imply $E(\epsilon|z)=0$. The model is nonparametric IV model with the structural ...
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Weak Law of Large Numbers: Conditional Expectations in Random Subsequences

Let $(X_i, Y_i)_{i=1}^{\infty}$ be iid continuous random vectors with continuous joint density, where $X_1$ have support $\mathcal{X}$. Let $B_n\subset \mathcal{X}\subset\mathbb{R}$ be decreasing ...
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Conditional Expectation Notation in ARCH Model

I'm new to ARCH models, and I have a question about the correct notation for expressing the conditional expectation of the return at time $t(r_t)$ given the information available up to time t-1. I'd ...
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Conditional Variance of $Z_i|\sum_i\beta_iZ_i$

Let's assume I have $K$ i.i.d. standard normal random variables $Z_1,...,Z_K$. Hence, I know that $V[Z_i] = 1$ and $E[Z_i] = 0$ for all $i\in K$. I am faced with computing the following conditional ...
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conditional expectation of univariate normal given realization of multivariate normal

Consider the random variable $\textbf{x} = (x_1, x_2, ..., x_N)$ where $x_i \sim N(\mu_i, \sigma_i)$ for $i=1,2,...,N$ and $\textbf{x} \sim N(\mu, \sigma)$ where $N$ stands for normal distribution, $\...
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Expected average distance in greedy matching on a circle

Now we have several independent and identically distributed random variables following the uniform distribution on the interval [0, 1].They are denoted as $x_1, x_2, x_3, ..., x_m$ and $y_1, y_2, ..., ...
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3 answers
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Scaling the conditioned random variable does not change conditional distribution, why?

Given two random variables $X$ and $Y$, I know intuitively that $$ \mathbb{E}[X\,|\,Y]=\mathbb{E}[X\,|\,cY], $$ where $c$ is some non-random constant. My intuition tells me that scaling the ...
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Understanding Fixed Regressors and Conditional Expectation on Fixed Regressors $E(Y|X_i)$

I'm having trouble with the statistical idea of a fixed regressor, it seems that our $X_i's$ are not treated as random variables, but we are still able to meaningfully condition $Y$ on them in a way ...
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Conditional expectation of Poisson, conditional on Poisson sums

Consider independent Poisson random variables $X_1\sim \text{Poisson}(\alpha_1)$, $X_2\sim \text{Poisson}(\alpha_2)$, $Y\sim \text{Poisson}(\lambda)$, and suppose $Z_1=X_1+Y$ and $Z_2=X_2+Y$. I want ...
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Interpretation of $\sigma$ in Gaussian mixture

I have a distribution of a variable that was normalized with plt.hist and then fitted with a sum of gaussian curves $g_M = \displaystyle\sum_i\frac{w_i}{\sigma_i \...
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BIvariate Normal and Conditional Expectation

I am working on a problem where I must show that the conditional distribution of Y given X follows the distribution with mean and variance shown below. In the previous question, we were given that X ...
Harry Lofi's user avatar
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The expected value and variance for the sum of 4 dice rolls only if a coin gives Head

Edited: toss a fair coin 4 times and then roll a fair 6-side dice whenever the coin gives a head H. Let X be the sum of the dice rolls. How to calculate E[X] and <...
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How should I best to use reported stats on the Tippy-top?

Suppose I have a large population, in the millions, drawn from some underlying distribution, which we will take as a member of a known distributional family with unknown parameters. Assume the ...
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Truncated Multivariate Normal expected value approximation

I have $\vec{x} \sim N(\vec{\mu}, \Sigma)$. I would like to calculate $$E[x_i | \vec{x} \geq 0]$$ There are libraries like tmvtnorm (in R) that calculates this for me. However, it seems to be very ...
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Optimal Conditional Distribution for Minimising Information-Theoretic Expression

Consider two countable sets $\mathcal{X}$ and $\mathcal{Y}$. I aim to find the conditional distribution $P_{Y|X}$ that minimizes the following expression for any $x \in \mathcal{X}$ $$\sum_y P_{Y|X}(y|...
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Asymptotic standard errors vs exact standard errors

I am getting confused about the derivation of standard errors for the OLS estimator $\widehat{\beta}$. I have seen two different ways to derive standard errors: (i) from the exact covariance matrix of ...
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Conditional expectation for doubly truncated bivariate normal distribution

The evaluation of the moments of doubly truncated bivariate normal distribution leads to the formulas with a great complexity. It has not been possible to derive explicit formulae for the moments ...
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Derivation of bias variance trade-off with or without conditional expectation?

I found this nice lecture here where the bias variance trade-off is explained using conditional expectation - using e.g. $E_{y|X}[...]$ In this lecture here I found another proof of the formula ...
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Is the effect in a Cox proportional hazard collapsible if the covariates are normally distributed and the baseline hazard is constant?

Since the Cox PH model is a non-linear model, we would expect the effect to be non-collapsible. i.e., the marginal and conditional effects differ. I did some calculation for a setting where the ...
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General Expression for the $t$-th difference of conditional means

In econometrics, it is common to work with the difference-in-differences of conditional means. For example, let $Y$ denote a variable of interest and $X_{1}$ and $X_{2}$ denote binary regressors. The ...
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Conditional variance formula for gaussian process classification

I am trying to understand the maths behind scikit learn's Gaussian process classifier. There is a link to the book from which the algorithm was taken. It is a bit involed and there is a particular ...
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Prove that the equality holds [closed]

How to prove that for any random variables $X$, $Y$ and $Z$ with finite variances, we have $Cov(X,Y)=E(Cov(X,Y|Z))+Cov(E(X|Z),E(Y|Z))$?
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Computing a conditional expectation function from data set

Say I have a two dimensional numerical data scatter $(x_i,y_i)$ corresponding to variables $x,y$, and I want to estimate the conditional expectation $\langle x|y\rangle$, what would be the procedure ...
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Conditional expectation function and causal inference

!For the question itself skip to the last paragraph! It is my understanding that iff we have a model of the form $$Y = m(X) + e$$ and $E[e|X] = 0$ we know that $m(X)$ is the conditional expectation ...
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In this RL problem, why is the substitution $q_*(A_t)=\mathbb{E}[R_t | A_t] \to R_t $ valid within this expectation (over actions)?

The question that follows is from a machine learning textbook (Reinforcement learning Suttion and Barto page 39 link). Given: a probability distribution over actions $x$ (a policy) at time $t$ ...
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Confused on Kullback-Leibler divergence being invoked without proper definition

I am trying to understand how authors of the DDPM paper in appendix A, made the leap from equation 21 to equation 22. Specifically, it is not clear to me how they managed to convert the first term of ...
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Definition of expectation with condition variables

I am having a hard time of digesting this, which is part of EM algorithm that I borrowed Equation 3.2.7 from https://www.informit.com/articles/article.aspx?p=363730&seqNum=2#:~:text=3.2%...
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$E[X|X^3-3X]=0$

Prove that $E[X|X^3-3X]=0$, with $f(x)=\frac{|x^2-1|}{4}$ being the density function of $X$ in the interval $[-2,2]$. My attempt: Let $Y=X^3-3X$ and $x_1, x_2, x_3$ the roots of $x^3-3x=y$. \begin{...
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Expected values conditioning two different expressions of a discrete variable

Suppose that we have two continuous random variables $Y$ and $X$ and a discrete random variable $W$. The discrete variable $W$ can have only three values 1, 2, or 3. That is, $Supp(W)=\left\{1,2,3\...
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An expression of a conditional expectation using the law of iterated expectation

Suppose that we have a discrete random variable $D$ with the support $\mathcal{S}_0=\left\{ d_1,\ldots,d_{20}\right\}$. In addition, consider a subset of the support like $\mathcal{S}_1=\left\{d_1,\...
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Give the distribution of an asymptotically Normal statistic, conditional on a function of sample ranks, and describe regression parameters

I'd like to write the following, about conditioning a normal r.v. on a rank statistic, but I'm unable to recall a specific theorem or theorems I can cite: "Let $\hat{\rho}$ be an asymptotically ...
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1 answer
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Measure Theoretic Explanation of Conditional Probability Given a Random Variable and Event

What does it mean in a rigorous measure theoretic sense to have the conditional probability of an event given a continuous random variable (or vector) and an event? As in, suppose $A,B$ are events and ...
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Conditional volatility and notation to express it

I'm taking a risk modelling class, and we're discussing conditional vs unconditional volatility. We have standard volatility $\sigma = \sqrt{\mathbb{E}[r_t - \mathbb{E}(r_t)]^2}$ But what is ...
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Measure Theoretic Justification For Manipulating Conditional Probability of Events Given a Continuous Random Variable and an Event [closed]

When considering the conditional probability of an event given a continuous random variable (or vector) can you essentially just manipulate the probability as if you were only working with discrete ...
PerpetuallyConfused's user avatar
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Checking the equivalence of conditional expectations

Let $X,X^1,\dots,X^{k-1},X^k\in\mathbb{R}$ be random variables, and let us define the conditional expectations as $$ f_{k-1}=\mathbb{E}[X|X^1,\dots,X^{k-1}]; \quad f_k=\mathbb{E}[X|X^1,\dots,X^k]. $$ ...
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Why is $P(A \mid C \cap B) = P(A \mid C)$ true in this instance?

As I was reading through this paper http://www.jstor.org/stable/25652278 I came across the following problem: Consider an urn with $N$ colored balls, the number of red balls, $X$, has a binomial ...
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Expected Error of OLS different from Zero

I am estimating a regression model for a dependent variable that could be grouped into 2 types based on characteristics that are not part of X. My out of sample forecasting errors are dependent on the ...
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Let $S = X + U, T = X + V$ for ind.normal r.v $X, U,V$. Is it possible to find $a, b$ real numbers such that $\mathbb{E}[X|S,T] = \mathbb{E}[X|aS+bT]$

Suppose we have random variables $$X \sim N(0, \sigma_X^2) \\ U \sim N(0, \sigma_U^2) \\ V \sim N(0, \sigma_V^2), $$ where $X$, $U$ and $V$ are independent. And $$ S = X + U \\ T = X + V. $$ Is it ...
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Group aggregates of conditional probabilities

I suspect I don't know the academic or SEO word for what I'm looking for. Put simply, I want to know a group's total expected success rate given each member's likelihood of attempting the "trial&...
Brandan's user avatar
6 votes
3 answers
353 views

Finding conditional expectation of conditional distribution

Let $a \sim N(\mu_a,1/\tau)$, and $s = a + \epsilon$, where $\epsilon \sim N(0,1/\eta)$. I know that because both $a$ and $\epsilon$ is normal distribution, s must also be normally distributed with $s ...
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Expected value of a variable that depends on two other random variables, and one of these random variables depends on another random variable

I am trying to solve the problem using conditional expectations. The expected value H is depends on the waiting time T and a set threshold X (a real number that is a constant random variable during ...
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