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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Conditional Remaining Time to Event Paradox
The conditional expected time remaining for an event to occur seems to grow with waiting time. This seems either wrong or like some sort of paradox.
Let's take the Lomax distribution as an example. ...
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Expectation of covariate conditional on regression, $E[X\mid Y=f(X,Z)]$
Suppose $X $ is a covariate in a regression model so $Y = f(X, Z) $ where $f$ is some (non-linear) link function and $Z$ is some other covariate. What is the $E[X\mid Y]$?
Would $E[X\mid Y] = E[X\mid ...
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Expectation of uniform distribution conditioned on an interval [duplicate]
I am trying to understand the concept of conditioning on an event better. To do so, I've cooked up the following toy problem then tried to generalize it in the context of uniform distribution.
Suppose ...
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Is conditional expectation evaluated by the copula strictly increasing when the correlation coefficient is positive and vice versa?
I used the copula to evaluate the $\mathbb{E}[Y|X]$ and from my experiments on some copulas, I observed that when the random variables have positive correlation coefficient, $\mathbb{E}[Y|X]$ is ...
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Diffusion Model consistency term derivation question
The consistency term of the diffusion model is written as:
$$\mathop{\mathbb{E_{q_\phi(x_{1:T}|x_0)}}} \left[\log\prod_{t=2}^T \frac{p(x_{t-1} | x_t)}{q_\phi(x_{t-1}|x_t, x_0)}\right]$$
$$= \sum_{t=2}^...
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Expectation of binomial random variable
Having trouble understanding something I read in a paper recently.
Say we have $X \sim \operatorname{Binomial}(N,p).$ The paper states:
$$E[X \mid N,p] = Np$$ (so far so good)
and
$$E[X] = \mu p$$
...
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How to evaluate this conditional expectation for the E-step in expectation-maximisation?
I'm trying to devise an expectation-maximisation algorithm for a certain problem but I'm unable to derive the conditional expectation in the E-step. For the purpose of this question I'll simplify the ...
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Posterior expectation of normal distribution with "truncated" observation
Consider the following problem of estimating an unknown parameter from normal samples:
Suppose that $\theta \sim N(0, \tau_\theta^{-1})$, where $\tau_\theta \ge 0$ is the prior precision.
Consider two ...
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$E[X\mid U,V]$ with dependent random variables [closed]
Let $(X,Y,Z)$ be a three-dimensional random variable with density function $f(x,y,z)=\frac{2}{3}(x+y+z)$ on $0<x,y,z<1$, $U=\min(X,Y,Z)$, $V=\max(X,Y,Z)$. Calculate $E[X\mid U,V]$.
I think the ...
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$E[X\mid \max(X,Y,Z)]$ with dependent variables
Let $(X,Y,Z)$ be a three-dimensional random variable with density function $f(x,y,z)=\frac{2}{3}(x+y+z)$ on $0<x,y,z<1$. Calculate $E[X\mid \max(X,Y,Z)]$.
I think the answer is $\frac{25}{36}\...
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How do you visualise the conditional esperance of your outcome variable depending on the predictor?
I've read several posts regarding the choice of distributions and link functions for GLMs. Although I'm far from understanding it all, what I've gathered is that the purpose of the link function is to ...
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Modelling the joint pmf of 2 correlated variables as p(x)*pmf(E(y|x))
Let x,y be 2 correlated counts. We want to model the joint pmf p(x,y). We know that p(x,y) = p(x)p(y|x) = p(y)(x|y). However, what happens when we don't know y|x, but we can estimate E(y|x)? Can't we ...
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Expectation conditional on a sigma algebra, what expectation does it refer to?
In a question like Intuition for Conditional Expectation of $\sigma$-algebra a concept like $E[X|\sigma(Y)]$ is used and I am puzzled about what sort of variable this actually is.
Say we have a ...
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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,\mathcal{A})\rightarrow(\mathcal{X}, \mathcal{F})$ and $Z:(\Omega, \mathcal{A}) \rightarrow (\mathcal{Z}, \mathcal{G})$ ...
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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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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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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 ...
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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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Showing $E[X\mid X^3-3X]=0$
Prove that $E[X\mid X^3-3X]=0$, 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 $x^3-3x=y$.
\...
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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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