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).
2
votes
0answers
13 views
Conditional mean and co-variance in $VAR(p)$ conditional on one lag only
Suppose I have a $p$'th order vector auto regression
$$\vec Z_t = F_1\vec Z_{t-1}+F_2\vec Z_{t-2} + \cdots +F_p \vec Z_{t - p}
+ \vec \epsilon_t,\qquad \vec\epsilon_t\sim N_q(\vec0,Q)$$
where $...
1
vote
0answers
37 views
Understand conditional expectation w.r.t. sigma -algebra [duplicate]
When a random variable $X$ is discrete, the definition of conditional expectation of $X$ with respect to a decomposition $\mathscr D$ is
$$
E[X|\mathscr D] = \sum_{i = 1}^m x_i \sum_{j = 1}^n P(X|\...
1
vote
1answer
35 views
How to deal with expected value in the context of time series?
For example, in this MA(2) model,
$y_t = u_t + \phi u_{t-2}$
$u_t$ is identically, independently, normally distributed with a mean of 0 and a variance of $\sigma^2$. (Does variance matter here?)
I ...
0
votes
0answers
34 views
Questions relating the definition of conditional expectation $E[g(X)|M]$ where is $X$ is random variable and M is an event
I saw the following definition of conditional expectation from a book:
if M is event and X is continuous random variable then we define:
$$E[X|M]=\int_{-\infty}^\infty xf(x|M)dx$$
Which is the ...
6
votes
1answer
113 views
Estimate E(Y|X,Z) from E(Y|X) and E[Y|Z]
Can I estimate $E[Y|X, Z]$ if I know $E[Y|X]$ and $E[Y|Z]$?
As an example, let's say I have a model where
$X\sim N(0, 1)$,
$Y = aX+\epsilon_Y, \epsilon_Y \sim N(0, 1)$,
$Z = bY+\epsilon_z, \...
0
votes
0answers
13 views
Calculate target variance contionaly to continuous variable
I'm trying to get the conditional variance of my target Y conditional to my inputs X. To do this I first discretized my inputs variable into 1 qualitativ variable and then I computed the variance of Y ...
2
votes
2answers
30 views
Generate Data where outcome is conditional on independent variables [closed]
I want to generate a synthetic dataset {Y, X1, X2}. Independent random variables X1 and X2 follow bernouli distribution where probabilities for X1 and X2 are known.
Whereas, outcome variable Y needs ...
-1
votes
1answer
22 views
Prove convergence in distribution, probability, or quadratic mean for a sequence of binary variables that depend on another binary variable
Suppose that $X$ has the support set $\{1, -1\}$, and $P(X = 1) = P(X = -1) = 0.5$.
Suppose that $X_n$ has the support set $\{X, e^n\}$, and
$P(X_n = X) = 1 - \frac{1}{n}$
$P(X_n = e^n) = \frac{1}...
2
votes
0answers
31 views
Proving covariance equals zero given a specific conditional expectation
I'm trying to prove the following:
Given $𝐸[𝑋|𝑌 = 𝛽] = 𝐸[𝑋]$ for any value of $\beta$, prove that $\operatorname{Cov}(𝑋,𝑌) = 0$;
So I was thinking to start with the definition of $\...
2
votes
0answers
67 views
Expected value of quotient of Poisson distributions
Let $X$ and $Y$ be independent random variables such that
$X \sim \text{Poisson}(\lambda \cdot c)$ and $Y \sim \text{Poisson}(\lambda \cdot (1-c))$, where $c$ is a real number in $[0, 1]$.
Is there ...
1
vote
0answers
24 views
Why can you not bypass the strong ignorability/unconfoundness assumption via iterated expectations?
Suppose we have that $\left(Y(1), Y(0)\right)$ are potential outcomes with $X$ being the covariate and $Z$ the treatment assignment. Typically in causal inference, one will assume strong ignorability ...
2
votes
1answer
53 views
Show $E[ (Y - E(Y|X)) (E(Y|X) - h(X))] = 0$
Show that $E[ (Y - E(Y|X)) (E(Y|X) - h(X))] = 0,$ where $X, Y$ are random variables with constant means and $h(x)$ is an arbitrary function.
So far, I have expanded out the expectation and used ...
13
votes
3answers
179 views
If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$
Question
If $X_1,\cdots,X_n \sim \mathcal{N}(\mu, 1)$ are IID, then compute $\mathbb{E}\left( X_1 \mid T \right)$, where $T = \sum_i X_i$.
Attempt: Please check if the below is correct.
Let ...
4
votes
1answer
125 views
Conditional Expectation of pdf
Wish to identify what I'm doing wrong when finding the $\operatorname E(X\mid Y=5)$ of the following:
$$f(x, y)=\begin{cases}
1/6 & \text{if } 0<x<2, 0<y<6-3x \\ 0 & \text{...
1
vote
0answers
14 views
Clarifying a proof of a particular paper on Steins Estimator
I am trying proving result (5.4) of the following paper. Its a paper on Steins estimator on spherically symmetric cases.
The doubt is a s follows:
Given
$$X|\theta\sim \mathcal{N}(\theta,I)$$ ...
2
votes
0answers
38 views
Linear Prediction and Linearity of CEF
I am revisiting the basic notions of linear regression and stumbled upon the following idea in Cameron and Trivedi's Microeconometrics book:
However, for the conditional mean to be linear in x, so ...
0
votes
1answer
22 views
Expectation of residuals in Zero Intercept Model
We know that the summation of residuals in a regression through origin model is not necessarily 0. Does that imply that Expectation of Residuals is not necessarily 0? CLRM still holds, so should that ...
2
votes
1answer
29 views
What does an expectation with respect to a policy mean in the reinforcement learning value function
I would like to know what the formal definition of the following expression is
$$
V_\pi(s) = \mathbb{E}_{\pi}(G_{t+1} | S_t =s)
$$
What does it mean to have the policy in the subscript? How would I ...
4
votes
1answer
92 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 ...
1
vote
1answer
71 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}^...
1
vote
0answers
36 views
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 :
$\Pr(Y,A,X)=\Pr(Y\mid A,X)\Pr(X\mid A)\Pr(A)$
$A \rightarrow (X,Y)$...
1
vote
1answer
38 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 ...
1
vote
1answer
21 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 ...
4
votes
1answer
39 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:
$$\mathbf{E}\left[\max\left(\sqrt{2}\left(\frac{\...
3
votes
2answers
160 views
Conditional probability for consecutive Bernoulli trials
Independent trials, each of which is a success with probability $p$, are performed until there are $k$ consecutive successes. Let $N_k$ denote the number of necessary trials to obtain $k$ consecutive
...
3
votes
1answer
53 views
Expected value conditional on a function
Let $X$ and $Y$ be random variables. What is the relationship (if any) between $E[Y|X]$ and $E[Y|g(X)]$?
I have been trying to Google or look in books but I'm having trouble even articulating this ...
4
votes
2answers
72 views
One-sided measurement error: $\widetilde{X} = X - \eta$, $\eta\geq0$. Is $E[X|\widetilde{X}=\widetilde{x}]$ increasing in $\widetilde{x}$?
Let $X\geq0$, $\eta\geq0$ and $X,\eta$ independent. We measure $X$ with a one-sided error: $\widetilde{X} = X - \eta$.
Is $E[X|\widetilde{X}=\widetilde{x}]$ increasing in $\widetilde{x}$?
0
votes
1answer
44 views
The expectation of the $K^{th}$ raw moment of $X$ given $X$: $E(X^k|X)$
Intuitively I believe $E(X^k|X)=X^k$, $k$ is a non-negative integer. One obvious special case is when $k=1$. Anyone has an idea how to prove it?
0
votes
1answer
19 views
Do discriminative models model conditional expectation?
In Machine Learning classic models like MLP, Logistic Regression or Linear Regression are called discriminative models.
I frequently read that those models estimate the conditional probability ...
4
votes
0answers
75 views
Taylor Series Expansion of Unconditional Expectation
We know that the best 1st order approximation of an unconditional expectation is the following-
$$E(y|x)=(E(y)-\beta E(x))+\beta x$$
where $\beta=\frac{\operatorname{Cov}(y,x)}{\operatorname{Var}(x)}...
0
votes
0answers
13 views
Conditional expectation of RVs under truncation-based stochastic signal release?
Suppose $z\sim N(0,1)$ and $x\sim N(\gamma z, 1)$, where $\gamma$ is a known sensitivity parameter. That is, $x=z+\epsilon, \epsilon \sim N(0,1)$ where $\epsilon$ is exogenous.
Further, suppose there'...
4
votes
3answers
72 views
Conditional expectation on an estimator for defensive sampling
In Introducing Monte Carlo Methods, by Robert and Casella, we have
How do we derive the second equality? Shouldn't it be
$$E\left[\frac{f(X_i)}{g_{Y_i}(X_i)}|X_i\right]=\frac{f(X_i)}{g_1(X_i)}\rho+...
2
votes
1answer
47 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
$1$...
1
vote
0answers
23 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 ...
4
votes
2answers
52 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.
0
votes
1answer
80 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=...
0
votes
0answers
3 views
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 ...
1
vote
0answers
39 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)]=\...
4
votes
2answers
83 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 ...
8
votes
2answers
99 views
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 ...
4
votes
1answer
64 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 ...
2
votes
2answers
212 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 ...
1
vote
0answers
9 views
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}...
0
votes
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 ...
1
vote
1answer
103 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 ...
1
vote
0answers
20 views
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 ...
1
vote
2answers
118 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 $\mu=0$ ...
1
vote
0answers
153 views
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) ...
0
votes
0answers
9 views
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:
$$
acc = \frac{\...
2
votes
1answer
33 views
Conditional expectation and variable decomposition
Suppose that $X$ and $Y$ have an uknown joint distribution $f_{XY}$.
How can I formally demostrate that it always exists a unique decomposition of the form :
$$ Y = E[Y|X] +\epsilon $$
without ...
