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)

Filter by
Sorted by
Tagged with
0
votes
1answer
23 views

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 ...
0
votes
0answers
30 views

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 ...
2
votes
1answer
46 views

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 ...
5
votes
1answer
60 views

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 ...
1
vote
0answers
204 views

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 ...
3
votes
0answers
32 views

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 ...
0
votes
0answers
10 views

Correct terminology to refer to a collection of conditional probability distributions

I would like your help to find the correct terminology to refer to a collection of conditional probability distributions. Consider two random variables $Y$ and $X$, with supports $\mathcal{Y}$ and $\...
3
votes
1answer
64 views

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 ...
0
votes
0answers
24 views

Implications of conditional mean independence

I have three random variables $Y,X,W$ with supports $\mathcal{X}, \mathcal{Y},\mathcal{W}$, respectively. I assume $E(Y|X,W)=0$ almost surely. Take two functions $z: \mathcal{X}\rightarrow \mathbb{...
0
votes
0answers
21 views

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? $$ \...
0
votes
1answer
44 views

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?
0
votes
1answer
73 views

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 ...
1
vote
1answer
212 views

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 ...
2
votes
1answer
49 views

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 ...
1
vote
0answers
67 views

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 ...
3
votes
2answers
65 views

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 ...
0
votes
0answers
78 views

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?
0
votes
0answers
25 views

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+...
4
votes
2answers
80 views

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. ...
1
vote
0answers
10 views

Is there an informative term for calling the random elements conditional on which a PDF of a random element is defined?

Let $X_{1}, \dots, X_{n}$ be i.i.d. random elements; suppose the conditional PDF $f_{X_{1} \mid X_{2} , \dots, X_{n}}$ exists. Then I wonder if there is already in literature an informative name for $...
4
votes
1answer
100 views

Is the parameter vector of an indentifiable distribution of a transformed random vector always a subvector…?

I would like, after further considerations about this problem, to reformulate this question of mine again. I kept a record of the past words and remarks as the appendix below. I think the question ...
2
votes
1answer
48 views

Question about sufficiency

I learned in my (classical) statistics class that (if we have densities) $T(X)$ is sufficient iff $$f(x)= g(T(x))h(x)$$ I am reading "the Bayesian Choice" and there the factorization-lemma is quoted ...
2
votes
0answers
28 views

Modelling resampling as conditioning on sum for independent discrete variables

I am trying to model a discrete data generating process where I first draw $Y = (y_1, ..., y_N), y_n \sim F(\theta_n)$ independently from some family of discrete distributions $F$ (e.g. negative ...
2
votes
1answer
54 views

Using posterior in an expectation

I am studying (myself, not in class) the book of Rogers & Girolami, A First Course in Machine Learning. In working through a logistic classifier, I found the equation $$ p(t_{new} = 1| \mathbf{...
4
votes
0answers
143 views

Distribution of conditional expectation?

Let $X,Y$ be random variables with pdf $f_{X,Y}$. I would like to find the distribution of the random variable $\mathbb{E}(Y\mid X)$, conditional expectation of $Y$ given $X$. If a specific form of $\...
1
vote
0answers
45 views

Conditional matrix normal distribution

Suppose $\epsilon$ is a $n\times p$ with independent rows $\epsilon_i\sim N(0, \Sigma)$. $Y$ is a matrix of size $n\times p_1$ and $X$ a matrix of size $n\times p_2$ constructed as $$ Y=XA+\epsilon B\\...
2
votes
0answers
62 views

Expressing the likelihood of the multivariate normal

I have two multivariate normal samples: $\boldsymbol{X^{(1)}=\left( \begin{matrix} \boldsymbol{Y^{(1)}}\\\boldsymbol{Z^{(1)}} \end{matrix}\right)}$ and $\boldsymbol{X^{(2)}=\left( \begin{matrix} \...
3
votes
1answer
397 views

Are normal prior and posterior random variables dependent?

Consider the usual normal Bayesian model. The prior $X=\mu_0+\sigma_0\epsilon_0$, where $\epsilon_0$ follows a standard normal distribution. The data $Y=X+\sigma_1\epsilon_1$, where $\epsilon_1$ ...
0
votes
1answer
56 views

notation: precedence of conditional when multiple variables

In expressions such as $ P(X,Y|Z) $ and $I(X; Y|Z)$ (mutual information) there are two interpretations for a student, and the correct one does not seem to be mentioned in textbooks. "joint ...
1
vote
0answers
195 views

How does conditional expectation relate to sufficiency?

In what follows, I will disregard all "measure-theoretic niceties about conditioning on measure-zero sets", as my professor calls it. I just want to know if the following general idea, or even just ...
2
votes
1answer
350 views

W is normal conditional on a normal variable Z. Does it necessarily follow that W is normal unconditionally?

$Z$ is normal with mean $\mu_z$ and standard deviation $\sigma^2_z$. Contitional on $Z = z$, $W$ is normal with mean $z$ and variance $\sigma^2_w$. Does it follow from these hypotheses that W is a ...
3
votes
1answer
550 views

UMVUE of location parameter (shifted exponential)

Let $X_1,...,X_n$ be a sample from a distribution with pdf, $f_X(x) = e^{-x + \theta}, x \geq \theta$. Let $x_0 \geq \theta$ be given. I'm trying to find the UMVUE of $f_X(x_0) = e^{-x_0 + \theta}$. I ...
5
votes
0answers
175 views

Simulate Gaussian variables conditional on their sum of squares

Consider a $d$-dimensional Gaussian random vector $\mathbf{Z}=[Z_i]_i$ with mean $\boldsymbol{\mu}$ and covariance matrix $\boldsymbol{\Sigma}$. What would be the more efficient method(s) to simulate $...
1
vote
1answer
102 views

Purpose of scatterplots

This is a beginner question, hopefully this is allowed here. I understand that scatterplots can be useful in showing the relationship between two variables, so I generated several plots of the ...
1
vote
1answer
551 views

Conditioning on continuous random variables

Conditioning is a tool I have used a lot in the discrete setting: usually this takes the form $$P(X=k) = \sum_{i=0}^\infty P(X=k | Y=i) P(Y=i).$$ I'm a little confused about the analogous situation ...
4
votes
1answer
178 views

Find the distribution of $Y-X|X=x$

The conditional distribution of $Y|X=x$ is Normal$(x,1)$. The marginal distribution of $X$ is Normal$(0,1)$ Find the distribution of $Y-X|X=x$. I notice that since the conditional distribution ...
6
votes
2answers
1k views

Sample random variables conditional on their sum

Let $(X_1, \dots, X_n)$ be an iid sample of random variables with a known continuous distribution. I would like to simulate such a sample, conditional on the value of its sum, that is: $$ X_1, \dots, ...
2
votes
1answer
468 views

What is the physical significance of cumulative correlation coefficient?

Say, I have 2 parameters, and based on my dataset, I have iteratively calculated the correlation coefficients between them by taking the correlation of the first i terms, where i ranges from 1 to the ...
2
votes
1answer
161 views

Investigate correlation conditional on a threshold

I have 3 variables in my data set. (i) My gut feel says variable1 and variable2 are correlated, only when variable3 >= threshold3. What is the technique I can use to see if this holds true, to ...
1
vote
1answer
115 views

Square of conditional variables

I thought that the following would hold but my classmates doubt me unfortunately: $Z = X|Y \rightarrow Z^2 = X^2|Y$ The original problem was an expectation of this and I tried to go back to the ...
4
votes
1answer
470 views

Non-parametric conditional variance estimation

Let $(x_i,y_i)_{1\leq i\leq n}$ some dataset. I want to estimate the conditional expectation $E[Y\mid X=x]$ and the conditional variance $V[Y\mid X=x]$. I used Nadaraya-Watson's estimator to estimate ...
9
votes
2answers
2k views

Conditional Distribution of uniform random variable given Order statistic

I have the following question at hand: Suppose $U,V$ are iid random variables following Unif$(0,1)$. what is the conditional distribution of $U$ given $Z:=\max(U,V)$ ? I tried writing $Z=\Bbb{I}\...
8
votes
1answer
403 views

Computation of Conditional Expectation on $\sigma$-algebras

I have not really seen any probability books calculate conditional expectation, except for $\sigma$-algebras generated by a discrete random variable. They simply state the existence of conditional ...
21
votes
3answers
2k views

Intuition for Conditional Expectation of $\sigma$-algebra

Let $(\Omega,\mathscr{F},\mu)$ be a probability space, given a random variable $\xi:\Omega \to \mathbb{R}$ and a $\sigma$-algebra $\mathscr{G}\subseteq \mathscr{F}$ we can construct a new random ...
3
votes
1answer
47 views

Conditioning within definition explanation

I have a doubt on the meaning of a conditioning within a definition. In a book I've found the following definition of upper tolerance limit: $P(P(X<\bar X+kS|\bar X, S)>p)=1-\alpha$ where $X$ ...
1
vote
1answer
162 views

Calculating percentile conditional on continuous variable

I have a collection of datasets with (x,y) - pairs where the value of y is conditional on x. Neither the distribution of x nor of y is known, n is ranging from 1000 to 10000. Here is the scatterplot ...
3
votes
2answers
245 views

Computing conditioned probability of $X$ by $U=\min(X,Y)$

Let $X$ and $Y$ be independent random variables with $P(X\leq x)=F_x(x)$ and $P(Y\leq y)=F_y(y)$. Let $U=\min(X,Y)$. I know that $F_u(u)=1-(1-F_x(u)(1-F_y(u))).$ By definition: $P(X \leq x |U=u)= \...
1
vote
2answers
167 views

P(X<Y|Z=t) where Z=min(X,Y)

Lets X and Y be uniform random variable where $x \in [0,a]$ and $y \in [0,b]$ where a < b. We design $Z=\min(X,Y)$. I know that the CDF of Z is $P(Z<z)=1-\frac{(a-z)(b-z)}{ab}$ And by ...
0
votes
0answers
119 views

Methodology to generate conditionally independent data correct?

I am trying to generate samples from continuous distributions that are conditionally independent. More specifically, I would like to generate samples from the following joint distribution $f(x,y,z)$ ...
1
vote
1answer
76 views

Implications of independence of several random variables

Consider 4 real-valued random variables $X,Y,Z,V$ defined on the same probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Assume that $X$ is independent of $Y,Z,V$, i.e. the probability ...