Questions tagged [conditioning]

Conditioning is a probabilistic operation that consists in examining the probabilistic properties of a random variable (or of an event) given the realised value of another random variable (or of an event)

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Where does the random come in for conditional expectations $\mathbb{E}[X | \mathcal{F}]$? [closed]

For continuous random variables $X, Y$ the conditional expectation $\mathbb{E}[X | Y]$ is itself a random variable. I understood this in the sense that for a realisation of $Y$ we can say $$ \mathbb{E}...
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how to view conditioning on a random variable rather than a particular value of that variable

The question is about $P(Y|X)$ versus $P(Y|X=x)$. Is there an alternate way to write $P(Y|X)$ that makes its meaning more clear? I believe these are correct equations for conditional entropy: $$ H(Y|...
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Condition on two random variables

I'm trying to set up the proper assumptions for a proof I'm working on: Given that $P(A|e) = P(A)$ and $P(A|c,e) = P(A|e)$, can we prove that $P(A|c)=P(A)$? I understand that A is independent of e and ...
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Correct conditional expectation with respect to a different measurable space

Suppose we've got random variables $X_1:(\Omega_1,\mathcal{A}_1)\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R})),X_2:(\Omega_2,\mathcal{A}_2)\rightarrow(\mathbb{R},\mathcal{B}(\mathbb{R}))$ on a ...
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Inference on a Gaussian random field / undirected graph?

Assume I have an undirected graph with $D$ nodes, and a $D$-by-$D$ matrix with edge strengths between $0$ (implying conditional indepedence given all other nodes), and $1$ (implying complete ...
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Conditional Tolerance Interval

With the generous help of people here, I've recently learned about the notion of a Tolerance Interval (Confidence in a range estimate, and +- 2sigma rule of thumb). I am now working on an application ...
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Probability Conditioned on Inequality

Assume that $A \sim \mathcal{N}(0, 1)$, $B \sim \mathcal{N}(0, 1)$. I am trying to calculate $P(A \,|\, A < B)$. For the sake of this problem, we can assume that $A \perp B$, but (for obvious ...
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Is it possible to calculate a bivariate normal density using only the univariate normal density function? [duplicate]

Suppose $(X,Y)$ are two jointly normally distributed random variables. Suppose further that we want to calculate the density of $(X=x,Y=y)$. Is it possible to calculate this density if we do not have ...
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If P(A) + P(B) = 1, does P(A|C) + P(B|C) = 1?

Just to explain where this is coming from. I was working on question 5(b) from stat 110 on conditional probabilities. I'll put a picture of the question and its solution below I worked on question 5(...
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What’s a textbook covering similar content to “Introduction to Probability Models” by Sheldon Ross?

I’m taking a class with a instructor using said textbook, and I find the explanations in it lacking. It’d be great if anyone can offer an alternative book covering similar content (i.e. conditioning ...
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How to find mean of multivariate normal distribution when holding a variable constant? [duplicate]

I wanted to know if there is a way to calculate the mean of a multivariate normal distribution when a certain variable is held constant. For example, if I had a continuous bivariate normal ...
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A regressor performs better under a certain regime -- how to condition that regressor to make regression better?

I ran a regression $Y \sim X_1 + X_2 + .. X_n$. I find out what one regressor , $X_1$'s performance (or correlation with $Y$) depends on another variable $t$ (not in the regression). So basically if I ...
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Variance of Bernoulli when success probability varies

Say the success probability $X$ is a random variable with mean $\mu$ and Variance $\sigma^2$ which takes values in $[0,1]$. How can I compute the variance of a random Variable $Y$ which is 1 with ...
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If $e_0$ are the OLS residuals, what is random in $\hat{\beta}_{OLS}|f(e_0) < \hat{\beta} < f^*(e_0)$?

This is a follow up question to the question I've posted here. Suppose $Y \sim N(X\beta, \sigma^2I)$, where $y \in \mathbb{R}^n$. Let $X \in \mathbb{R}^{n \times p}$ denote a full rank design matrix. ...
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Test for conditional dependence

I am trying to find out whether two variables are conditionally dependent on each other. I would like to have a test for B implies A (B->A) or not and one for A implies B (A->B) or not. It is ...
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Find conditional PMF of a multinomial distribution

I am trying to find the conditional PMF of a multinomial analytically and though I know my result is wrong I can't seem to pinpoint where my argument is wrong. Seeking help to find my mistake. Given: $...
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Distribution of R|R+T for R,T binomials with the same probability

$$R \sim \mathrm{Bin}(N_R, p)$$ $$T \sim \mathrm{Bin}(N_T, p)$$ What is the distribution of $R$ given particular value of $R + T$? My guess would be that $R | R + T \sim \mathrm{Bin}(R + T, {N_R \over ...
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Gaussian Processes as weighted averages?

I've been wondering if a "weighted average" is a valid means to consider the Gaussian Process, specifically in the context of GP Regression. The kernel (I'll be referring to the common ...
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Expected value of max of two discrete random variables

I'm reading this paper An Efficient PTAS for Stochastic Load Balancing with Poisson Jobs. Which is solving a makespan minimizing job-shop problem for Poisson job sizes. Basically, schedule the minimum ...
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Markov transition matrix row sums to 1

I am trying to learn a little bit of Markov Chains through Dobrow's "Introduction to Stochastic Processes with R", but i am struggling with the following: The entries of every Markov ...
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conditional expection gaussian vector

I have got an question on computing conditional expection I was working on the following conditional expectation problem find $$\mathbb{E}[X-Y|2X+Y]$$ where $\begin{bmatrix}X \\ Y \end{bmatrix} \sim \...
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How conditioning happens?

F. Elwert and C. Winship in the paper "Endogenous Selection Bias: The Problem of Conditioning on a Collider Variable" (content available here) discuss conditioning on different types of ...
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Observational study comparing 2 products with different (but overlapping) feature coverage

I have two software products $A$ and $B$ which form treatment $X$. $B$ is a new version of $A$ and in development. So $B$ does not have all the features that $A$ has. However, $B$ is designed to make ...
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Graphical representation of unconditional expected value

First, let tell you that I've being struggling with the concept of unconditional expectation for linear regression. For conditional expectation is easier: We know that the conditional expectation ...
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Covariance of X and Y conditional on X+Y>Z? [closed]

Suppose that $X$, $Y$, and $Z$ are three independent random variables. Is there a way to compute the following conditional covariance? $Cov(X, Y | X + Y \geq Z)$
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How to notate joint conditional probability

Given $X = \{ x_1, x_2, \dots, \}$ and $Y = \{ y_1, y_2, \dots \}$ let $P(X,Y)$ be their joint probability. Conditioning $P(X, Y)$ on $y \in Y$ corresponds to looking at the distribution of the ...
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Definition of covariate-specific effect: why after, not before intervention?

Pearl et al. "Causal Inference in Statistics: A Primer" (2016) p. 70 contains the following text regarding conditional interventions and covariate-specific effects: [S]uppose a doctor ...
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Gibbs sampling for Multivariate: how to update?

In this page of Murphy's 'Machine Learning: a Probabilistic Perspective' it's explained how to do Gibbs sampling on a Gaussian Mixture Model. Reading this, I was trying to understand when to update ...
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Typo in Deepleariningbook.org or am I misunderstanding Bayesian stats?

This is on page 133 of the book: https://www.deeplearningbook.org/contents/ml.html#pf10 In the above, it says that the data set is directly observed and so is not random If that data we observe ...
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generating process of acceptance-rejection algorithm

The acceptance-rejection algorithm is described as follows: suppose you have RVs $X$ and $Y$ with densities $f_X$ and $f_Y$, respectively, and there exists a constant $c$ such that $\frac{f_X(t)}{f_Y(...
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7 votes
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Is there a difference in interpretation between $Y|X = m(X) + \epsilon$ vs. $Y = m(X) + \epsilon$?

I understand that $E(Y|X)$ and $E(Y)$ are different, but difference sources, when $Y$ is a function of other random variables such as $X$, use $Y|X$ and $Y$ to describe this relationship. I'm not sure ...
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Why condition on either the r.v. $X$ or $Y$ and integrate over a product of pdfs rather a single pdf to find this probability density?

Let $X$ have the probability density $f_{X}(x)=\lambda e^{-\lambda x}, \;\; x>0$ and let $Y$ have the probability density $f_{Y}(y)=\lambda e^{-\lambda x},\;\; y>0.$ Find the probability ...
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14 votes
1 answer
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What does the assumption of the Fisher test that "The row and column totals should be fixed" mean?

As this source states, one of the assumptions to perform Fisher's exact test of independence is that the row and column totals should be fixed. However, I find the explanation coming with it pretty ...
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conditioning for deep neural networks

What is the best way to do conditioning when working with deep neural nets? For example, say we we want to condition a VAE on the class i.e. CVAE. There exists different ways of adding the class to ...
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What is E[X|Y>c] where Y=X+Z, and X, Z are normally distributed rvs? [duplicate]

Suppose X, Z are normally distributed random variables and independent ($X$ follows $N(\mu,\sigma^{2})$ and $Z$ follows $N(0,\sigma_{z}^{2})$), and that Y=X+Z. What is E[X|Y>c], where c is just a ...
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2 votes
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Find conditional pdf given joint

Let the joint pdf of $X$ and $Y$ be $f(x,y) = 12e^{-4x-3y}, x>0, y>0$. What is the marginal cdf of $X$? of $Y$? Am I just supposed to integrate f(x,y) with respect to $x$ or $y$ to get the ...
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Meaning of "for each value $X = x$, the random variable Y can be represented in the form $Y = \beta_0 + \beta_1 x + \epsilon$" in linear regression

In what follows assume $Y: \ \Omega \to \mathbb R$ and $X: \ \Omega \to E$ The following quoute is from page 700 in DeGroot and Schervish - Probability and statistics, introducing a simple linear ...
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When can we use fixed design regression results for the random design setting? [closed]

Suppose I have an independent vector $X$ and a dependent scalar random variable $Y$ and I wish to construct a regression model to predict $Y$ using $X$ given data $\{(x_i,y_i)\}_{i=1}^{n}$. For ...
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3 votes
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What does "Normal distribution conditioned on $x \mod 1$" mean?

I understand what $Pr[X = x | Y = y]$ means, however, on this paper (Lemma 10 in Appendix A), we have the following: Let $D_r$ denote the continuous Gaussian distribution of parameter $r$,i.e., the ...
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3 votes
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Consistent estimator for conditional expectation

Take sequence of random vectors $(Y_i, X_i)_{i=1}^N$ i.i.d. $X_i$ has finite support. Let $x$ be a point in the support of $X_i$. Consider $E(Y_i|X_i=x)$. Suppose it exists and is finite. Is it ...
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Conditioning and linear MSE

Let $\sigma_{X|Y}^2$ denote the linear mean squared error in estimating $X$ from $Y$. Then is it always true that additional conditioning cannot increase the LLSE? In other words, is this true? $$ \...
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How time series structure can affect the independence of residuals condition for MLR?

I am going through all four conditions for Multiple Linear Regression and stick with this question: what happens with the independence if we have time series data structure?
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Computing Variances by Conditioning

I have trouble with the first part of this problem (Please take a look at the image below). This is an example problem from my old textbook years ago and I have had trouble understanding: How Y is ...
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1 vote
1 answer
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Law of Iterated Expectations Example

Consider a randomized experiment (AB test), where $n$ units are randomized into the treatment group $T_i=1$ and control group $T_i=0$. Let $M_i\in P$ denote the observed value of a continuous variable ...
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Is the conditional distribution of Y given X the most we can know about how X "affects" Y?

In his book "Introductory Econometrics", Jeffrey Woolridge states "The most we can know about how X affects Y is contained in the conditional distribution of Y given X". Is this statement true? Would ...
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RiskMetrics VAR calculations and conditional distribution of sum of log returns

According to Tsay's book in Chapter 7, for the Risk Metrics model: A nice property of such a special random-walk IGARCH model is that the conditional distribution of a multiperiod return is ...
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How can I estimate the confidence interval of correlations possibly dependend with time?

I have a multivariate problem (with solar data from different meteorological stations) that I am working on my engineering master thesis. I would like to estimate the correlations of different ...
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Does conditioning on a random variable yield a random variable?

Let $X$ and $Y$ be random variables and $y\in Im(Y)$ a possible value of $Y$. Is $X|Y=y$ a random variable in the mathematical sense? Or is that just abusing notation?
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Defining "variance" of a partially defined random variable

Elsewhere within CrossValidated the following survey sampling problem was mention. To each member $i$ of a population $\{1,\ldots,N\}$ there is assigned some value $c_i$ whose average $\mu=(c_1+\cdots+...
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5 votes
2 answers
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Bayesian statistical conclusions: we implicitly condition on the known values of any covariates, $x$

My Bayesian data analysis textbook says the following: Bayesian statistical conclusions about a parameter $\theta$, or unobserved data $\tilde{y}$, are made in terms of probability statements. ...
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