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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Expected Mean using Total Law of Expectation

Given a coin having probability of p of coming up heads is successively tossed until two of the most recent n tosses are heads where n >= 2. Let N be the number of tosses. Determine E[N] So I have ...
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28 views

Is it true that $\mathbb{E}_B[\mathbb{E}_{A|B} [g(A,B) | B]]=\mathbb{E}_A[\mathbb{E}_{B|A}[g(A,B)|A]]$?

Take a function $g$ of two random variable $A$, $B$ which are not necessarily independent. Is the following true: $$ \mathbb{E}_B\big[\mathbb{E}_{A|B} [g(A,B) | B]\big] = \mathbb{E}_A\big[\mathbb{E}_{...
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Is this true:$ E[A|C]=E[A|B∩C]P(!B|C)+E[A|!B∩C]P(!B|C)$?

That is the conditional extension of $$E[A]=E[A|B]P(B)+E[A|!B]P(!B)$$ $$E[A|C]=E[A|B∩C]P(B|C)+E[A|!B∩C]P(!B|C)$$ allowed me to get the right answer. Thank you for the help.
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Why there is only a single model coefficient in mixed models, if it's conditional to random effects?

I have a mixed model, where client ID is included as a random effect. It is a random intercept and slope model. I was told that mixed models have different interpretation than marginal models, and the ...
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Show that $\mathbb{E}(P_s|P_s) = P_s$ [duplicate]

Let $(P_t)_{t \geq 0}$ be a Poisson process with $\lambda > 0$. I want to show that $\mathbb{E}(P_s|P_s) = P_s$. I compute $$\mathbb{E}(P_s|P_s) = \sum_{x \in \text{Img}(P_s)} x \cdot \mathbb{P}(...
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1answer
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Expectation of A given A,B Random variables

So I am self-teaching myself some stuff about random variables and expectations for a course that I am going to take in the upcoming semester, and I found some resources online for properties that I ...
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How can I calculate Conditional expectation using copula

Let X, Y two time series and $F_{i, \beta_i}$ the marginal distribution of residual of each time series and beta is vector of their parameter. I studied the dependence between this two series using ...
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When to use Monte Carlo test type in ctree?

I'm a user of ctree function from partykit package in R. I always wondered for which purpose we want to use Monte Carlo to compute the distribution of test statistics? The literature suggest that it ...
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Finding conditional expectation: $\mathrm{E}(aX + b | cX + d < eY + g)$

I'm trying to obtain a formula for the following object, without imposing any distributional assumptions: $$ E(aX + b | cX + d < eY + g) $$ Obviously by linearity of $E$, $\ aE(X| \frac{eY + g - b}...
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marginal conditional from joint of three r.v.'s

I have two random variables, $X$ and $Y$. I know that $X \sim \text{Gamma}(a,b)$ and $Y \sim \text{Gamma}(c,d)$. Furthermore, I know that $Z \sim \text{Poisson}(XY)$. I know the joint distribution ...
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Show difference between conditional expectations is positive

Suppose we have a continuous random variable $Y$ and a random Bernoulli variable $T$ such that $P(T=1|Y)$ is monotonically increasing in $Y$. How can we show that $E[Y|T=1]>E[Y|T=0]$? To me, it ...
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Why Condition on a Random Variable in Iterated Expectation?

Say I wanted to take the following expectation for $C$ a constant and an RV $\theta$ with density $p(\theta)$. $$E[\theta-C] = \int_{\theta}(\theta-C)p(\theta)d\theta$$ Now given $p(y|\theta)$ and a ...
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Conditional mean of Weibull to the power of N

How can I (is it possible to) derive the conditional expectation of a variable $w$ that follows a 2 parameter Weibull Distribution $W(\lambda,k)$ with $\lambda$ scale parameter and $k$ shape parameter?...
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Proof of propensity score theorem - problem with conditional expectation

My question is about the "propensity score theorem", which is quickly presented here : https://en.wikipedia.org/wiki/Propensity_score_matching#Main_theorems The paper of the authors of the theorem, ...
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MCMC inside Expectation Maximization

I wish to optimize the following likelihood function for parameter $\Theta$: $$p(D|\Theta)=\int_X\int_Y p(x, y, D|\Theta)dydx$$ where $X$ and $Y$ are latent variables and only $D$ is observed. I ...
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Variance of expected value, is the formula right?

In this video and this video, I am seeing the variances of expected values calculated as this: and this: From which, I derived the formula: $$\displaystyle\textrm{var}\big(\mathbf E[X\mid Y] \big) =...
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Conditional expectation of linear regression estimators

I'd like to clarify the steps to find the conditional expectation of two different estimators. My answers differ from the solution provided. The model provided is $y_i=x_i\beta+u_i$ & $E(u_i|x_i)...
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Sequential testing of hypothesis

Let $\{X_n : n ≥ 1\}$ be a sequence of i.i.d random variables with common density $f_\theta(\cdot )$. Lets we want to test \begin{eqnarray} \begin{array}{cc} H_0: & \theta=\theta_0 \\ H_1: & ...
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Rewriting the mutual information of a linear model through conditional expectation

I am reading the following paper: http://web.mit.edu/18.325/www/telatar_capacity.pdf In this paper we have the following linear model, with $\mathbf{n}$ being additive noise: \begin{equation} \...
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Exponential distribution memoryless question [closed]

Can someone explain why E[Y] = 0.9 would imply the probability in the solution?
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Can we express an AR(1) process as follows?

If $X_t$ follows an AR(1) process as follows \begin{equation} X_t=\rho X_{t-1}+\varepsilon_t \end{equation} Would it be correct to express the above as \begin{equation} X_t=\mathbb{E}\{X_t\mid X_{t-...
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What's an example where the expectation of product is zero but not conditional mean?

I am studying linear regressions. In this business, sometimes we can prove the results we want with the assumption that the error term $U$ is such that $E(XU) = 0$. But for lots of other results, in ...
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Conditional Expectation given the following equations

Suppose you have a Log-Normal model with: $\log\left(x_t\right) = \mu_t + \varepsilon_t$ $\mu_{t+1} = \mu_t + \delta_t$ $\delta_{t+1} = \delta_t + \gamma_t$ where $\varepsilon_t \sim \text{NID}\...
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If $E(Y|X_1=x_1,X_2=x_2)=0$ and $Var(Y|X_1=x_1,X_2=x_2)=C$ for all possible $x_1,x_2$, then what is $E(Y|X_1=x_1)$ and $Var(Y|X_1=x_1)$? [closed]

Given random variables $Y,X_1,X_2$, if $E(Y|X_1=x_1,X_2=x_2)=0$ and $Var(Y|X_1=x_1,X_2=x_2)=C$ (where $C$ is a constant) for all possible combinations of $x_1$ and $x_2$, then is $E(Y|X_1=x_1) = 0$ ...
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1answer
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Expecation of Linear Regression Coefficients

Let the entity ${\widehat{\boldsymbol\beta}}$ be a linear estimator (not necessarily the least squares estimator) of the true coefficient ${{\boldsymbol\beta}}$ in the regression of 𝐲 on 𝐗. In this ...
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Prove or disprove the Linearity of Expectiles used in Expectile Regression

For expectation (mean), there are many useful properties such as Linearity of Expectation: $\mathbb{E}[X+Y]=\mathbb{E}[X]+\mathbb{E}[Y]$ $\mathbb{E}[\alpha X]=\alpha\mathbb{E}[X]$ (The 2 equations ...
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1answer
127 views

Conditional expectation for maximum function

I have a discrete-time Markov chain queuing problem. Packets (computer packets, that is) arrive in the intervals. $A_n$ denotes the number of arrivals in the interval $(n - 1, n)$, where $n \ge 1$, ...
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1answer
59 views

Counterfactual Expectation Calculation

$\newcommand{\doop}{\operatorname{do}}$ Problem: (This is from Study question 4.3.1 from Causal Inference in Statistics: A Primer, by Pearl, Glymour, and Jewell.) Consider the causal model in the ...
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1answer
33 views

Minimising Posterior expected loss

I am new to Bayesian statistics. I am not sure how I could find the best point estimator $T(x)$ for $\alpha$ that minimises the posterior expected loss, $$E_{\alpha|x} [L(\alpha, T(x)))] = \int L(\...
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34 views

How to calculate conditional expectation of two random variables

How do I calculate E{Y|Z}? I know I have to consider the case for Z = 1 and Z = 0 but im not sure how to formulate the conditional pdf then make the calculation. What are the bounds the the ...
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How to compute a conditional expectation with respect to a sigma algebra that is not generated by a partition?

This question is not asking for a general algorithm. Instead, a very specific, preferably simple, example would suffice. Similar questions have been asked and answered for sigma-algebras that are ...
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1answer
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What is a satiated model, and how is it non-linear?

I am just beginning to learn econometrics, and am a little confused by my lecture notes. They say that the conditional mean function has a known functional form, and is linear in parameter, e.g.,: $$...
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How to use Lemma 5.1 to get Equation (5) in infoGAN paper

Could someone please offer a proof that how to apply lemma 5.1 to get eq.5 in the infoGAN paper. The lemma is as follows, Lemma 5.1 For random variables X, Y and function f(x, y) under suitable ...
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When using OLS on $\ln(y) = \beta_1 \ln(x) + \epsilon$, is $\beta_1$ the elasticity of $E[y\vert x]$, or the $y$ in the data (or both)?

Specifically, suppose we are estimating $$ \ln(y)=\beta_1\ln(x) + \epsilon $$ I understand that $\beta_1 = \frac{\partial \ln(y)}{\partial \ln(x)}$ which is the elasticity of $y$ with respect to $x$ ...
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1answer
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Doubt in conditional hope. Uniform conditioned to a bernoulli

I'm having a hard time getting conditional hope. I know that $$P(X = x | Y = y) = y^x(1-y)^{1-x}, \:\: x = \{0,1\}, \:\: 0 \leq y \leq 1.$$ Besides that, $Y \sim U[0, 1]$. I want to get $E(Y|X=x).$ ...
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How $E(X/X+Y)=E(Y/X+Y)$ when $X,Y$ are i.i.d's [closed]

How $E(X/X+Y)=E(Y/X+Y)$ when $X,Y$ are i.i.d's I have recently started studying probability and statistics on my own. I am presently watching harvard lectures. In that professor told that the above ...
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1answer
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Value of Regression Coefficient of Conditionally Independet Variable

Suppose we known that $E[Y\mid X=x]=f(\alpha + x\beta)$ for some pararmetic prediction method $f$. Consider now $E[Y\mid X=x, Z=z]=f(\alpha + x\beta + z\gamma)$. If $Y \perp\!\!\!\perp Z\mid X$ under ...
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Conditional expectation of random variables defined off of each other

First of all, when we say that $X_n \sim \text{Unif}(0,X_{n-1})$, what does that mean, rigorously? Does it mean that for every $\omega \in \Omega$, $X_n(\omega)\sim \text{Unif}(0,X_{n-1}(\omega))$? ...
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Mean and variance of conditional distribution of truncated normal distribution

Let's say that I have $$\begin{pmatrix} x \\ y \end{pmatrix} \sim N \begin{pmatrix} \begin{pmatrix} \hat{x} \\ \hat{y} \end{pmatrix}, \begin{pmatrix} A & C \\ C^{T} & B \end{pmatrix} \...
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1answer
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Conditional Variances Question

I am doing a question out of Statistical Inference by Casella and Berger, (4.58c). I am having trouble with the brief solution that my teacher discussed. $X$ and $Y$ are random variables with ...
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1answer
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OLS as approximation for non-linear function

Assume a non-linear regression model \begin{align} \mathbb E[y \lvert x] &= m(x,\theta) \\ y &= m(x,\theta) + \varepsilon, \end{align} with $\varepsilon := y - m(x,\theta)$...
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Taylor expansion of a conditional expectation of a function of a random variable

I saw this post on hazard function and conditional pdf, Why is the Hazard function not a pdf?, and the main outcome there is that the argument of a conditional pdf cannot depend on the conditioning ...
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Conditional mean of MVN random vector

Suppose $\mathbf{r} \sim \mathcal{N}_n(\mathbf{\mu}, \mathbf{\Sigma})$. How can I work out (or estimate numerically) the mean vector for $\mathbf{r}$ conditional on $\sum_{i=1}^n c_i \frac{r_i}{r_i + ...
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Expected value $\mathbb{E}[X-Y \mid X>Y, Y< c]$ for independent variables $X,Y$?

I am looking for the expected value $\mathbb{E}[X-Y \mid X>Y, Y< c]$ for independent variables $X \sim U(x_1, x_2),Y \sim U(y_1,y_2)$, $c\in [y_1, y_2]$? Is the following formula correct? $$\...
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1answer
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elementary coin-toss problem

Toss a coin 10 times. Find the expected number of heads in the first 5 tosses, given 6 heads in the 10 tosses The given solution: Think of a box containing 6 heads and 4 tails. Draw 5. The ...
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Is $E[g(X_1,…,X_n)] = E[E(g(X_1,…,X_n) | X_2,…X_n)]$?

I know that $E(X_1) = E[E(X_1 | X_2)]$, but I’m wondering if I can generalize this to $E[g(X_1,…,X_n)] = E[E(g(X_1,…,X_n) | X_2,…X_n)]$ based on the following: $$E(g(X_1,…,X_n)) = \int_{-\infty}^{\...
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1answer
63 views

can zero covariance and zero expectation imply zero conditional expectation?

$x$ and $\epsilon$ are two random variables. If $Cov(x, \epsilon)=0$ and $E[\epsilon]=0$, can that lead to $E[\epsilon|x]=0?$
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1answer
71 views

How to solve for conditional expectations with binary random variables?

Suppose you have a binary random variable X with a probability of 50% 1, and a probability of 50% 0. Another random variable Y has the conditional expectation E[Y|X]=5. Another random variable W=XY-X. ...
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32 views

Estimating Survival Time Given A Set of Data Points for CDF

Suppose I have a set of data points to describe a cdf(cumulative density function) for X: for example, (cumulative probability, x): (0.3, 100), (0.5, 200), (0.9, 400), (0.95, 800), (0.99, 1000). How ...
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1answer
25 views

basic conditional expectation

Toss a die 10 times. If you get six 1's, find the expected number of 2's. The answer given is $\frac{4}{5}$. I'm trying to understand where the following method of solving the problem goes wrong. $...

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