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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Is it true that if a parameter enters into a conditional PDF then it must also enter into the joint PDF?
Let $(x,y) \sim f_{x,y}$ be a random vector in $\mathbb{R}^{2}$. If there is exactly one $b \in \mathbb{R}$ such that $y = xb + u$ and $\mathbb{E}(u \mid x) = 0$, then the conditional PDF $f_{y \mid ...
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44 views
How to calculate the integral of Normal CDF and Normal PDF?
I'm trying to find $\int_{\frac{a-b}{B}}^\infty\Phi\left(tA+ABx\right)\phi(x)\,dx$ where
$A = \frac{\sqrt{\gamma_{3}+\sigma_3^2}}{\gamma_{3}},\ B = \frac{\gamma_{2}}{\sqrt{\gamma_{2}+\sigma_{2}^...
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When to use all 3 variables in a graph to estimate a conditional expectation of 2
The title might not be perfect. But here goes:
Suppose there are 3 variables $(A,X,Y)$. And they have the following dependencies :
$Pr(Y,A,X)=Pr(Y\mid A,X)Pr(X\mid A)Pr(A)$
$A \rightarrow (X,Y)$
$...
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1answer
28 views
Conditional expectation of Poisson process given number of events
Let $\{N(t), t\geq 0\}$ be a Poisson process with rate $\lambda$, $S_n$ the instant of the $n$-th arrival and $T_n$ the $n$-th interarrival time, that is, $T_n = S_n - S_{n-1}$, $n \geq 1$.
Now ...
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1answer
13 views
Simple Appplication of Law of Iterated Expectation
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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1answer
30 views
Integral from the Adversarial Spheres paper (maximum of the difference between a constant and a normal random variable)
I'm trying to follow a proof in the Adversarial Spheres preprint on arXiv. The proof requires the computation of the integral in Appendix F, page 14:
$$\mathbf{E}\left[\max\left(\sqrt{2}\left(\frac{\...
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141 views
Conditional probability for consecutive Bernoulli trials
Independent trials, each of which is a success with probability $p$, are performed until there are $k$ consecutive successes. Let $N_k$ denote the number of necessary trials to obtain $k$ consecutive
...
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Assess expected performance on trials with different parameters?
I work at an online retailer, and we sell our products by sending out push notifications to specific places about specific products. A particular notification is called an offer. I'm trying to assess ...
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1answer
46 views
Expected value conditional on a function
Let $X$ and $Y$ be random variables. What is the relationship (if any) between $E[Y|X]$ and $E[Y|g(X)]$?
I have been trying to Google or look in books but I'm having trouble even articulating this ...
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72 views
One-sided measurement error: $\widetilde{X} = X - \eta$, $\eta\geq0$. Is $E[X|\widetilde{X}=\widetilde{x}]$ increasing in $\widetilde{x}$?
Let $X\geq0$, $\eta\geq0$ and $X,\eta$ independent. We measure $X$ with a one-sided error: $\widetilde{X} = X - \eta$.
Is $E[X|\widetilde{X}=\widetilde{x}]$ increasing in $\widetilde{x}$?
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37 views
The expectation of the $K^{th}$ raw moment of $X$ given $X$: $E(X^k|X)$
Intuitively I believe $E(X^k|X)=X^k$, $k$ is a non-negative integer. One obvious special case is when $k=1$. Anyone has an idea how to prove it?
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1answer
14 views
Do discriminative models model conditional expectation?
In Machine Learning classic models like MLP, Logistic Regression or Linear Regression are called discriminative models.
I frequently read that those models estimate the conditional probability ...
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Taylor Series Expansion of Unconditional Expectation
We know that the best 1st order approximation of an unconditional expectation is the following-
$$E(y|x)=(E(y)-\beta E(x))+\beta x$$
where $\beta=\frac{\operatorname{Cov}(y,x)}{\operatorname{Var}(x)}...
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Conditional expectation of RVs under truncation-based stochastic signal release?
Suppose $z\sim N(0,1)$ and $x\sim N(\gamma z, 1)$, where $\gamma$ is a known sensitivity parameter. That is, $x=z+\epsilon, \epsilon \sim N(0,1)$ where $\epsilon$ is exogenous.
Further, suppose there'...
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10 views
Conditional Expectation of generalized function
Can anyone help me to find a solution
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We study the diffusion of a message (say, like a tweet) on a social network. To this end, we use the following simplified model. Let Xn be ...
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Conditional expectation on an estimator for defensive sampling
In Introducing Monte Carlo Methods, by Robert and Casella, we have
How do we derive the second equality? Shouldn't it be
$$E\left[\frac{f(X_i)}{g_{Y_i}(X_i)}|X_i\right]=\frac{f(X_i)}{g_1(X_i)}\rho+...
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1answer
43 views
Average treatment effect in binary choice model
All the random variables below are defined on the same probabiluty space $(\Omega, \mathcal{F}, \mathbb{P})$.
Consider the following model
$$
Y\equiv 1\{\epsilon > \beta_0+\beta_1X\}
$$
where
$1$...
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22 views
Are these two ways of looking at conditional linear regression the same?
Let $X$, $\hat{S}$, $\hat{V}$ be random variables, with means (and conditional means, if necessary) $=0$.
Is $\beta^*(X) = \frac{\mathrm{Cov}(\hat{S}, \hat{V}|X)}{\mathrm{Var}(\hat{S}|X)}$ the ...
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2answers
36 views
Does this equality hold?
Is it true that
$E[XY|Z]=E[X|Z]E[Y|Z]$ if X and Y are independent each other, but X,Y are not independent with Z? Can anyone prove this? Thank you
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1answer
58 views
Missing steps in Bellman Equation and MDP assumptions
My question is similar to this one. I am trying to fill in the missing steps of the Bellman equation for a Markov Decision Process:
I will focus on the first term of the sum $R_{t+1} + \gamma \sum_{k=...
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Compare health spending per capita of 5 countries conditional on their age structure, health index and GDP per capita
I am interested to answer the following question: How does the health spending per capita in 2010 of 5 countries different if they have the same age structure in their population, health index and GDP ...
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35 views
Conditional distribution relations
There is a probability density function of the form,
$f_S(s)=\displaystyle\iint f_S(s|x,y)f_{X,Y}(x,y)dxdy$
that is used for evaluation of expectation of some monotonic function
$\mathbb{E}[g(S)]=\...
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2answers
79 views
Conditional expectation given event and random variable
Let $W$ be a random variable that only takes on the values $1$ or $0$. Let $X$ and $Y$ be two other random variables. I came across the following:
$$\mathbb{E}(Y|W=1, X)$$
How is this 'conditional ...
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Ms. A selects a number $X$ randomly from the uniform distribution on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers
Ms. A selects a number $X$ randomly from the uniform distribution
on $[0, 1]$. Then Mr. B repeatedly, and independently, draws numbers
$Y_1, Y_2, ...$ from the uniform distribution on $[0, 1]$, until ...
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1answer
59 views
Calculating conditional expectation and mean time to failure
I was reading text on probability where they state:
$$\operatorname{Ex}[C]=\sum_{i=1}^{\infty}i\cdot\Pr[C=i]=\sum_{i=0}^{\infty}\Pr[C>i]$$
Now assuming there is a system which fails at each step ...
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2answers
113 views
UMVUE for Bernoulli
Let $X_1,..,X_n$ be independent and $Bin(1,\theta)$ distributed. I would like to find the UMVUE for $\phi(\theta)=\theta^3$. I have a complete and sufficient statistic in $T=\sum_iX_i$, and a unbiased ...
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Conditional Expected Value of Birnbaum-Saunders Bivariate
Consider $(T{1},T_{2})\sim BVBS(\alpha_{1},\beta_{1},\alpha_{2},\beta_{2}).$ According to item (b) of Theorem 3.1 of Balakrishnan and Kundu (2010) article, we have:
\begin{align*}
f_{T_{1}|T_{2}=t_{2}...
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1answer
31 views
Construct joint distribution of $X,Y$ such that $E[X|Y=y,y\geq \bar{y}]$ is piecewise linear
Can one construct a joint density $f(x,y)$ such that the marginal distribution of $Y\sim~U[c,d]$, no restrictions on $X$ (it would be great that $X$ also has uniform distribution) as long as it has ...
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1answer
102 views
How to improve an estimator for a Poisson sample
Given the statistical model $(\mathbb N_0^n, P(\mathbb N_0^n),\operatorname{Poi}(\vartheta)^{\otimes n}:\vartheta >0)$, $T(X)=X_1X_2$ is an unbiased estimator of
$\vartheta^2$. I want to improve ...
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Standard errors of OLS estimate if regressor is a stochast?
Assume the model classical linear regression model (with for simplicity only one regressor)
$$y=X\beta +u,$$
with $u$, $X$ independent, and $\operatorname{Var}(u|X)=\sigma^2I_n$. Assume for ...
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2answers
107 views
Expected value of $X$ which follows a normal distribution, between a certain interval [duplicate]
What is the process of finding the expected value of $X$ in a normal distribution between a certain interval?
In particular I want to find: $E(X | a \le X \le b)$.
For example, if $X$ has $\mu=0$ ...
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Multiplicative or additive error model in linear regression
Suppose that $Y$ and $X$ have an unknown joint distribution with an arbitrarly complicated unknown C.E.F. $ \; E[Y|X]=\mu (X)$.
Suppose that we want to find the best (in MSE minimization terms) ...
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accuracy conditional on feature values
I have a binary classification model and I would like to estimate the accuracy as a function of another variable.
To be clearer, I can compute the usual accuracy on the testset:
$$
acc = \frac{\...
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1answer
26 views
Conditional expectation and variable decomposition
Suppose that $X$ and $Y$ have an uknown joint distribution $f_{XY}$.
How can I formally demostrate that it always exists a unique decomposition of the form :
$$ Y = E[Y|X] +\epsilon $$
without ...
2
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0answers
38 views
Model assumption redundancy on simple linear regression
I'm stuck on some subtleties involved in simple linear regression formulation.
In a very general fashion I understood that the optimal point forecast for a random variable $Y$ (in terms of MSE ...
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1answer
47 views
Conditional Expectation of a Random Forest Model
Suppose $y = f(x_1,...x_n)$, and a random forest model $\hat{f_{\mathrm{rf}}}$ is constructed so that $\hat{y}_1 = \hat{f_{\mathrm{rf}}}(x_1,...,x_n)$. In this case, $\hat{y}_1$ is an estimate of $\...
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2answers
94 views
Expectation of $X_t^2$ in GARCH(1,1) model
Given a GARCH(1,1) model
\begin{aligned}
X_{t} &= \sigma_{t}Z_{t} \\
\sigma^{2}_{t} &= \omega + \alpha_{1}X^{2}_{t-1} + \beta_{1}\sigma^{2}_{t-1}
\end{aligned}
with $Z_{t} \sim i.i.d(0,1)$:
...
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1answer
62 views
Closed form Conditional Expectation of Gamma Distributed Variable
I am trying to solve a missing values problem using EM algorithm and I am having trouble deriving closed form solution in the Expectation step. Say we have gamma distributed variables $X^{(1)}...X^{(...
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1answer
85 views
Monte Carlo Method for approximating conditional expectation
I would like to compute E[X | X > a] for X ~ Gamma(.) using a Monte Carlo (simulation) apporoach. Any idea how I would go about it?
I have a method but I am not sure if it's correct or consistent:
<...
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Conditional expectation of the probability that a prior parameter is greater than some value
I am working in a Bayesian setting where I have a prior $p \sim \text{Beta}(\alpha, \beta)$. For reasons that don't really matter, I'm later defining a new parameter, call it $C$, which is the ...
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42 views
Law of Large Numbers Convergence for Different Means
Considering the LLN stating that:
If we have $\{X_i\}$ $iid$ then as $n \rightarrow \infty$ it follows
that $\bar{X} \rightarrow \mu = E[X_i]$, almost surely
In my case, we have a logistic ...
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80 views
Expected Value of a Poisson distribution given the mean is a Gamma distributed RV
An insurance company supposes that the number of accidents that each of its policyholders will have this year is Poisson distributed, with a mean depending on the policyholder: the Poisson mean Λ of a ...
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65 views
For random vectors $X = (X_1, \ldots, X_m)^T \in R^m$ and $Y = (Y_1, \ldots, Y_n)^T \in R^n$
Here is an exercise on linear transformations of random vectors. I don't understand why we have to have a condition on $X$. If anyone can help me with these properties, that would be great.
For ...
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Proving $P(X=Y)=1$ given that $E(Y|X)=X$ and $E(X|Y)=Y$
It's a Prove/Disprove question.
Given $\mathrm{E}(Y|X)=X$ and $\mathrm{E}(X|Y)=Y$ and both $\mathrm{E}(X^2)$ and $\mathrm{E}(Y^2)$ are finite, then $$P(X=Y)=1$$
If we somehow get $\mathrm{Var}(X-Y)...
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1answer
232 views
Why does the conditional expectation minimize L2 loss? [duplicate]
The claim is that, for a regression task, the conditional regression function $f(x) = \mathbb{E}[\mathbf{Y}|\mathbf{X}=x]$ minimizes the L2 loss $\arg\min(\mathbb{E}[\mathbf{Y} - f(\mathbf{X})]^2)$. I ...
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2answers
99 views
What is the expected value of $X$ given $X<Y$, where $X,Y\sim\mathcal{N}(\mu,\sigma^2)$? [duplicate]
What is $\mathbb{E}[X|X<Y]$ if $X,Y\overset{iid}{\sim}\mathcal{N}(\mu,\sigma^2)$?
I have found that $\mathbb{E}[X|X<Y]=\int_{-\infty}^{\infty} -\log(\Phi(\frac{x-\mu}{\sigma}))x\phi(\frac{x-\mu}...
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0answers
31 views
How to generate data that have given conditional mean and conditional quantile using R?
Suppose I want to generate independent data $(y_{i},x_{i})$ such that the conditional mean of $y_{i}$ given $x_{i}$ is a quadratic function in $x_{i}$ and the $.25$ conditional quantile of $y_{i}$ ...
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1answer
128 views
Technical point about convergence with conditional expectation
I have a sequence of non negative variables $X_n$ such as:
$$E(X_n|C_n)=\frac{C_n}{n^2}$$
where $C_n$ is a sequence of random variables converging almost surely to $1$.
Can I conclude $X_n$ tends to ...
2
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0answers
40 views
Truncated mulitvariate normal: first two moments
Let $X\in \mathbb{R}$ be a univariate random varible for which it holds that $$ X \sim N(\mu,\sigma^2).$$
where $\mu\in \mathbb{R}$ gives the expected value and $\sigma^2>0$ is the variance.
If ...
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1answer
22 views
Efficient estimation of conditional means from pdf, CDF, & quantile function supplied numerically
Suppose I have a a probability distribution that I know to have a continuous differentiable unimodal pdf, with pdf(x) strictly greater than zero for all x in the positive half-plane. In addition, I ...
