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 expectation with updated information
Let
$$\epsilon_{t+1} = \rho\epsilon_t + \eta_{t+1}$$
$$E_t[r_{t+k}|\eta_t] = \phi^k \eta_t$$
Can we say that
$$E_t[r_{t+1}] = \sum_{k=0}^\infty \phi^k \eta_{t-k}$$
Are there any conditions for this ...
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1answer
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Deriving reward functions in Sutton & Barto
Does anyone know how the equations have been derived, I'm still learning probablity theory and expectations
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Truncated expectation of sum of independent random variables
Take three random variables $X$, $Y$, $Z$ s.t. $E[X]>0$, $E[Y|X]=0$, $Z = X+Y$.
What can I say about $E[x| x> k]$ vs. $E[z| z>k]$ where $k>0$? Intuitively, the latter should be bigger ...
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Conditional expectation of functions of two random variables with inequality conditions
Consider general case first. Let $X$ and $Y$ be independent continuous random variables with known pdfs. What is expectation of
$Z = \begin{cases}
g_1(X, Y), & Y \geq X,\\
g_2(X, Y), & Y < ...
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1answer
35 views
Where is the error?
I am trying to compute expectation of $X\mathbb I_{[X+Y\le a]}$ where $a$ is a fixed positive integer, $X$ is discrete uniform random variable taking values from $1$ to $a$, and $Y$ another random ...
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Correlation and expected values
Consider two random variables, $x$ and $y$. Denote the correlation between them by $\rho$. Assume that $E[x]$ is also a function of some parameter $\pi$ and is increasing in $\pi$. So if we increase $\...
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1answer
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Can we always write a random variable as conditional expectation plus error?
Consider the random variables $Y,X$. I believe that we can always write
$$
Y=E(Y|X)+\epsilon
$$
with $E(\epsilon|X)=0$.
Question: Is the above true regardless whether $Y$ is a discrete or continuous ...
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1answer
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Derivation of expected loss ESL (integrating over conditional expectation confusion)
I am trying to understand the derivation of expected loss (equation 2.11 in Elements of Statistical learning) and there is a specific step I do not understand.
We start with
$EPE(f) = E(Y - f(x))^{2}$
...
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conditional expectation of normal r.v [duplicate]
Let $X$ and $Y$ i.i.d standardized normally distributed random variables.
Calculate the conditional expectation of :
$$ \mathbb{E}[(X+Y)^{3} | \mathscr{G}] $$
where $\mathscr{G} = \sigma(X)$ ($\sigma$...
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31 views
cauchy schwarz inequality on sum of squares
Can I and how I can use the Cauchy Schwarz inequality on the amplitude of a imaginary sum of squares?
$$Z = X+iY$$ and $$|Z| = \sqrt{X^2 +Y^2}$$
to show that $$|E[Z] | \leq \mathbb{E}[|Z|]$$
where $X,...
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1answer
51 views
conditional expectation of iid $X,Y$ cubic sum
Let $X$ and $Y$ i.i.d standardized normally distributed random variables.
Calculate the conditional expectation of :
$$ \mathbb{E}[(X+Y)^{3} | \mathscr{G}] $$
where $\mathscr{G} = \sigma(X)$ ($\sigma$...
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0answers
14 views
Conditional Probability involving condition on two RVs
Suppose X,Y~exp(2) and are independent. Let W=X+Y.
How do I set up integrals to calculate the following:
f(W|X>Y)
E(W|X>Y)
Thanks!
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Optimality of multivariate conditional expectation
It is well known result [1] that among all functions of random variable $Z$, the conditional expectation is the unique minimizer of the expected Bregman loss i.e.,
$$ \mathtt{arg \quad min}_{Y \in \...
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Prediction In Generalized Linear Models and Factorized Conditional Expectation
I have been introduced to generalized linear models as follows: Let there be sample data $x_1,\dots,x_n\in\mathbb{R}^p$ of the r.v. $\mathcal{X}$ organized as rows into the matrix $X$, and $y\in\...
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1answer
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How does $E(u)=\int E(u|v)p(v)dv$?
I'm reading Bayesian Data Analysis: http://www.stat.columbia.edu/~gelman/book/
and this equation was given
$$E(u)=\int E(u|v)p(v)dv$$
Can someone explain this to me? I don't see how they're equivalent,...
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1answer
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Dimension changes when taking conditional expectation
I am trying to derive a formula of a paper, and quite stuck at how they did that. Hopefully, someone can help me. So as we all know, for a $d$-dimensional random vector $\boldsymbol{X} \sim \mathcal{N}...
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Custom distribution function
I am looking to build a two-parameter distribution function $f\left(x\mid m,v\right)$, potentially with bounded support. For this distribution, the expected value only depends on $m$ and the variance ...
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1answer
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The formula for conditional expectation in terms of joint cdf
We know that covariance can be written as a function of marginals and joint CDFs, namely
$$\newcommand{\cov}{\operatorname{cov}}\newcommand{\d}{\mathrm{d}}\cov(X,Y) = \iint (F_{X,Y}(x,y) - F_X(x)F_Y(y)...
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How to understand $E(X) = E(X|\mathcal{D})P(\mathcal{D}) + E(X|\mathcal{D}^c)P(\mathcal{D}^c)$ when $P(\mathcal{D}) = 0$
Suppose $Y \in \mathbb{R}^n$ and $Z \in \mathbb{R}^n$ are random vectors, where $Y$ follows a $MVN(\mu, \Sigma)$ distribution. Let $X \in \mathbb{R}^{n \times p}$ be a full-rank fixed matrix of ...
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1answer
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Assuming $y \sim N(X\beta, \sigma^2I)$, and $e = y - X\hat{\beta}$ are residuals, what does $E(\hat{\beta}|e = e_0)$ mean?
Let $X$ be an $n\times p$ full-rank matrix of predictors with $n > p$, and $y$ be the vector of responses. Assume $y \sim N(X\beta, \sigma^2I)$, and let $\hat{\beta}$ be the least-squares estimator ...
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If $\mathcal{A}$ is a union of disjoint sets $\mathcal{A}_i$, is $E(X|\mathcal{A}) = \sum_{i=1}^k E(X|\mathcal{A}_i)$?
Suppose $X$ is a random variable. Let $\mathcal{A}$ be a set that is made up of disjoint subsets $\mathcal{A}_1, \ldots, \mathcal{A}_k$, i.e.
$$\bigcup_{i=1}^k \mathcal{A}_i = \mathcal{A}, \quad \...
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1answer
61 views
Computing probability distributions in the two envelope problem
I am trying to understand the resolution to the two envelope problem. While I am still working my way through it and so far the progress has been good I am stuck at a claim that one of the sources ...
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29 views
Zero conditional expectation implying zero covariance?
Proof: E[X|Y]=0 implies COV[X,Y]=0
I was thinking maybe the law of total covariance or tower rule but couldn't come up with the proof
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1answer
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If $X, Y$ are random vectors, is it true that $E[g(X,Y)] = E[E[g(X,Y)| \{Y: AY \geq b\}]]$?
Suppose $Y \in \mathbb{R}^n$ and $X \in \mathbb{R}^n$ are random vectors. By the law of iterated expectation, does the following hold?
$$E[g(X,Y)] = E[E[g(X,Y)| \{Y: AY \geq b\}]],$$
where $g(\cdot, \...
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1answer
40 views
Expected value of a Gumbel variable conditional on Gumbel being the maximum of N iid Gumbel
I found the following results in Hanemann (1984) which I cannot find a proof for. I checked through simulation that it is right, but I would like to see an analytical proof...
Hanemann, W. M. (1984). ...
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1answer
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The average treatment effect and the difference in means
Hi I have a question related to the treatment effect.
Recently, I am reading literatures on treatment effect and have a question.
In the literatures, we denote the counterfactual outcomes as $Y_1$ and ...
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1answer
47 views
Conditional bias-variance decomposition of MSE
The MSE can be decomposed as follows:
\begin{align*}
\mathbb{E}\left[(\hat{\theta} - \theta)^2\right] &= \mathbb{E}\left[\left(\hat{\theta} - \mathbb{E}(\hat{\theta}) + \mathbb{E}(\hat{\theta}) - \...
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Monte-Carlo Estimation of conditional expectation term
I want to ask if my approach to estimation of the following quantity is correct:
I have $n$ i.i.d. draws $\{(X_i,Z_i) \}_{i=1}^n$ and I want to estimate for a fixed $(i,j)$ pair the quantity:
$$
\...
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1answer
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The conditional normal distribution [duplicate]
I would like to find the conditional bivariate normal distribution.
There are two dependent normal variables with the same distribution and the correlation coefficient $\rho$: $X,Y \sim N(\mu, \sigma^...
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Do the following equations for the conditional expectations hold under the given assumptions?
Let $(\Omega,F,\mathbb{F}=(F_n)_{n\in\mathbb{N_0}},P)$ be a filtered probability space and $(Z_n)_{n\in\mathbb{N_0}}$ be a sequence of integrable iid random variables. Denote $F_n=\sigma(Z_o,\dots,Z_n)...
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2answers
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When is the conditional mean of potential outcomes linear in the propensity score?
Consider an outcome $Y$, treatment $D$, and set of covariates $X$. The outcome is real-valued, the treatment has support $\{0,1\}$, and the set of covariates is a collection of binary variables with ...
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1answer
42 views
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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2answers
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Interpretation of multivariate conditional gaussian function form?
I've been reading over this Multivariate Gaussian conditional proof, trying to make sense of how the mean and variance of a gaussian conditional was derived. I've come to accept that unless I allocate ...
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The expected value of waiting time
A two-way street (i.e. vehicles pass from left and right). The number of cars coming from the left side follows a Poisson distribution with $\lambda$= 20 cars per minute, and the number of cars coming ...
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1answer
82 views
conditional expectation: an implication
$E[X|Y]$ is equal to the almost surely unique and deterministic function of
$Y$, say $\varphi (Y)$, such that
$$E[X f(Y)]= E[\varphi (X)f(Y)]$$
for all bounded, measurable and non-negative $f$'s.
How ...
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1answer
101 views
Conditional Expectation Function in linear regression
I've already read this: Conditional expectation function
For this question, we assume familiar notation in linear regression, with $Y$ being the response and stochastic regressors $X$. I've seen both $...
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0answers
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Calculate Conditional Expectation using dataset and covariance matrices
So I'm having trouble understanding what I'm doing wrong here.
For context, I have some velocity components in my dataset for turbulence (simplified).
I have flattened them out so my 3 velocity ...
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0answers
58 views
Unbiasedness of estimators of conditional expectation with discrete dependent variable
I'm trying to figure out whether the basic formula for a conditional expectation with discrete conditioning variable (let's call it $X$).
The basic argument can go something like the following: a ...
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Independency vs ignorance - why is conditioning on a variable interpreted as ignoring the other one? (partial dependency plots)
Setting
Let $f(X)=f(X_S, X_C)$ be an estimator of $y$ using random variables (or random vectors) $X_S$ and $X_C$. The authors (see below) mention that:
$$ \tilde{f}_S (X_S) = \mathrm{E}( f(X_S, X_C) | ...
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Expectation Maximization Correctness of Problem Formulation
Suppose I draw $n$ iid samples from a Poisson$(\lambda)$ distribution, with $\lambda$ unknown. Now, I artificially turn every 3 I draw into a 1, so that the probability of observing any particular non-...
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Why is $E_t[g|x]$ the notation of $\int_t g(x,t)p(t|x)dt$?
Why is $E_t[g|x]$ the notation of $\int_t g(x,t)p(t|x)dt$?
This is seen in the equation(1) of the first answer of this question Loss functions for regression proof
This notation is also seen in the ...
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1answer
81 views
Rao Blackwell theorem on Bernoulli distribution
I need help with the following Problem: Let $X_1,...,X_n$ be a random sample of iid random variables,
$X_i\sim Ber(p), p\in (0,1)$. We want to estimate $\theta = p^2$. It is known, that $\hat{\theta}(...
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Bivariate normal and truncated expectation [duplicate]
What is the expectation
$$\mathbb E[X_1 \lvert X_1 > X_2]$$
assuming that
$$(X_1,X_2) \sim \mathcal MVN(0,\Sigma),$$
with $\mathcal{MVN}$ being the multivarite normal.
I would expect this to have ...
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1answer
31 views
What's the relation between the conditional expectation of the score of the likelihood, and score of the conditional likelihood?
We wish to estimate the parameter $\boldsymbol\theta$ from data $\bf X$ in the presence of nuisance parameter $\boldsymbol\Phi$.
Suppose $\bf T$ is a complete and sufficient statistic for $\boldsymbol\...
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Why do I get $E[Y^2] > 0$ with one approach but $E[Y^2] = 0$ with another approach for $Y|X \sim N(0,\sigma_X^2)$?
Suppose $Y|X \sim N(0,\sigma_X^2)$ where $\sigma_X^2$ is the variance of $X$. Does this mean $Y$ itself has a normal distribution? Because it seems like this would cause a problem. If $Y$ has a normal ...
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1answer
63 views
Prove that $E(X) = P(X<a)E(X|X<a) + P(X\geq a)E(X|X\geq A)$
Question
While reading a Wikipedia article on Markov's Inequality, I came across the statement
$$E(X) = P(X<a)E(X|X<a) + P(X \geq a)E(X|X\geq A)$$
In the context of Markov's inequality, we are ...
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5answers
185 views
confusion about individual notation
Let's say I am trying to estimate the regression of $y$ on $x$:
$$y= x \beta + \epsilon.$$
So, when moving to regression frameworks, I often see people use the individual notation:
$$y_i = x_i \beta + ...
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0answers
11 views
Condition Random Variable on Range of Another Random Variable [duplicate]
Assume that $v_s \sim N(\mu_s,\sigma_s^2)$ and $v_b \sim N(\mu_b,\sigma_b^2)$, denote their correlation by $\rho$, and assume they are jointly normally distributed. How would I assess $E[v_b|v_s\leq c]...
3
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1answer
73 views
Conditional expectaction with probabilities for a sum of independent random variable
I have a r.v $S_N$ built as a sum of Bernoulli with parameter $p$. So $S_N = X_1 + X_2 + \ldots + X_N$. There is a second variable N, such that $N \sim Poisson(\lambda) $.
I have to compute:
$P(S_N=0)...
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1answer
19 views
Convergence of the variance of the posterior expectation
Consider a classical Bayesian model :
$$ \begin{array}{cc}
\theta \sim \pi \\
X = (X_1, ..., X_n) \overset{i.i.d.}{\sim} \pi(.\mid\theta)
\end{array}
$$
where the prior does $\pi$ and the likelihood $\...
