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Questions tagged [conditional-expectation]

For every question related to the concept of conditional expectation of a random variable with respect to a $\sigma$-algebra. It should be used with the tag (probability-theory) or (probability), and other ones if needed.

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Finding the conditional expectation of independent exponential random variables

Let $X$ and $Y$ be independent exponential random variables with respective rates $\lambda$ and $\mu$. Let $M = \text{min}(X,Y)$. Find (a) $E(MX|M=X)$ (b) $E(MX|M=Y)$ (c) Cov$(X,M)$ (a) I first ...
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How could $E(X\mid Y)$ be a function of $Y$?

When I was solving $ \operatorname{Cov}(X,E(X\mid Y)) = \operatorname{var}(E(X\mid Y))$, I notice that $E(X\mid Y)$ was treated as a function of $Y$. My thinking is $E(X\mid Y)$ is taking values of $ ...
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Prove that $E[x | s, y] = b_0 + b_1 s + b_2 y$ for some constants $b_0$, $b_1$, and $b_2$.

Let $E[z | v]$ denote the conditional expectation of the random variable $z$ conditional on the random variable $v$. Assume that $s = x + \epsilon$, where $\epsilon$ is a random normal variable ...
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19 views

Expectation property of martingales

Suppose $(Y_n)_{n=0}^\infty$ is a martingale of discrete random variables. Put $A:= \{(Y_0, \dots, Y_{n-1}) = (y_0, \dots, y_{n-1})\}$ In a proof I'm reading, it is claimed that $$\mathbb{E}[...
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40 views

Conditional Expectation of Continuous Random Variable

Let $Y$ be a random variable of density g. How could I compute the expectation $E[Y|Y<a]$ ? Thank you in advance!
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31 views

Conditional Variances: show that $Var(X-E(X|Y)) = Var(X|Y)$ [on hold]

How do I show that for two random variables $X$ and $Y$ $$ Var(X-E(X|Y)) = Var(X|Y) $$
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Conditional expectation for a simple random walk

Suppose that $S_n$ is a simple random walk started at $0$, so that $S_n = X_1 + \dots + X_n$ where $X_j$'s are iid random variables taking values $1$ and $-1$ with probability $p = 3/4$ and $q=1/4$ ...
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Why are these two definitions of conditional expectation equivalent?

From Rick Durrett's book Probability: Theory and Examples: We define the conditional expectation of $X$ given $\mathcal{G}$, $E(X | \mathcal{G})$ to be any random variable $Y$ that has (1) $Y \...
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Conditional expectation problem with integration limit and not only

We are in the shopping center. Customer is paying with card with chance $60\%$, and witch cash with probability $40\%$. When client is paying with card the time of service has exponential distribution ...
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Double Expectation over Two Samples of 2 Joint Variables

I have 2 random variables X and Y. These variables are joint in the sense that $y_t = f(x_t) + v_t$ with $v_t$ i.i.d zero mean and finite covariance. We have two independent samples (x,y) and $(\hat{x}...
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How to compute a conditional expectation

I want to compute a conditionnal expectation, i know that $Z=(Z_1,\ldots,Z_p)'$ where $ Z_j=\Phi ^{-1}(U_j)$ with $Z \sim N(0,R(\theta))$ and $R(\theta)$ the $p \times p$ positive definite ...
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Conditional expectation property proof: $E(E(Z\mid X,Y)\mid X)=E(Z\mid X)$

$E(E(Z\mid X,Y)\mid X)=E(Z\mid X)$ The conditional expectation $E(Y\mid X)$ is defined as the (almost surely unique) function $\phi(X)$ such that $E(\phi(X)h(X))=E(Yh(X))$ for any bounded function $h$...
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Conditional expectation for multivariate normal distribution

Assume $\mathbf{X}=[X_1,X_2,X_3]$ follows a multivariate normal distribution, $\mathbf{X}\sim N_3 (\mu,\Sigma)$, where $\text{cov}(X_i,X_j)\neq 0$, for $i\neq j$. What is $\mathbb{E}[X_1 X_2|a_1X_1+...
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48 views

If $E[X|Y= y] = y^2$ for all $y$, does it follow that $E[X| Y] = Y^2$?

Let $X,Y: \Omega \to \mathbb{R}$ be two discrete random variables on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$, and $\mathbb{E}[|X|]< \infty$. Suppose that $\mathbb{E}[X|Y=y] = y^2$ ...
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Conditional expectation in the discrete case

Let $X: \Omega \to \mathbb{R}$ be a random variable with $\mathbb{E}[|X|]< \infty$, and suppose that $Y: \Omega \to \mathbb{R}^d$ is a discrete random vector. Define $$f(y) = \begin{cases}\frac{1}{...
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1answer
20 views

Substitute Concrete Value in Conditional Expectation

Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a probability space. Let $$ X, Y : \Omega \rightarrow \mathbb{R} $$ be random variables. Furthermore, let $$ f: \mathbb{R}^2 \rightarrow \mathbb{R} $$ be a $...
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51 views

Correlation between $X$ and $Y$ conditioned on $A\cup B$

Suppose $\theta \sim \text{Uniform}[0,2\pi]$, and $X=\cos(\theta)\,,\, Y=\sin(\theta)$. Let two events be defined by $A=\{X\geq 0\}\,,\, B=\{Y\geq 0\}$. I want to find the correlation coefficient ...
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56 views

Expected Solution of a Stochastic Differential Equation as a Conditional Expectation (this is a tough one).

On all you geniusses out there: this is a tough one. Preliminaries and Rigorous Technical Framework Let $T \in (0, \infty)$ be fixed. Let $d \in \mathbb{N}_{\geq 1}$ be fixed. Let $$(\Omega, \...
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27 views

Expectation of a random variable given another random variable

Under a group insurance policy, an insurer agrees to pay all of the medical bills incurred during the year by employees of a small company, up to a maximum total of one million dollars. The total ...
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44 views

Conditional expectation of random variable given minimal sigma-algebra generators

I am trying to understand better the definition of conditional expectation, for that I want to prove the following: Let $X$ be a random variable in the probability space $(\Omega, \Sigma, \mathbb{P})$...
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Finding $E(X\mid1_{\{X+Y=0\}})$ where $X,Y$ are i.i.d Bernoulli variables

I have two independent random variables $X,Y$ with the same following distributions : $$P(X=0)=P(Y=0)=1-p\,,\, P(X=1)=P(Y=1)=p$$ I want to calculate $E(X\mid 1_{\{X+Y=0\}})$. So Let's define ...
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Probability theory: If $X|Y=p $ is a first success distribution with parameter $p$ and $Y \in U(0,1)$. What's the distribution of X?

I've tried using approximations for expected values of functions and varaince of function. But it doesn't seem to add up to something I recognize. I think I'm doing something wrong. Should I maybe ...
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1answer
34 views

If $X_n$ is a submartingale, and $M,N$ are stopping times with $M \leq N$, then $X_M \leq E(X_{N} \mid \mathcal{F}_M)$

I'm trying to show that if $X_n$ is a submartingale, with stopping times $M \leq N$, $P(N\leq k) =1$, then $E(X_{N} \mid \mathcal{F}_M) \geq X_M$. The hint given is to use that, for any $A \in \...
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Stitching up conditional expectations

I want to say that if we have different partitions of some random variable, and we want to "stitch" them up to figure out the expected value of this random variable, the appropriate way to do so is to ...
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6 views

Conditional Expectation Property on two sub-sigma fields with no further assumptions on the fields

I was looking on some old notes on probability theory and I found the following "Let $\left(\Omega, \Sigma, P\right)$ be some probability space, $\Sigma_1, \Sigma_2 \subseteq \Sigma$ be two sigma ...
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Problem with counting conditional expectation with PDF

Let $(X,Y)$ be a random variable with PDF : $$f(x,y)=\begin{cases} e^{-x-y} &if\;x,y\ge0 \\ 0&in \;other\;case \end{cases}$$ I want to calculate $E(X+Y|X<Y).$ $E(X+Y|X<Y)=E(X|X<Y)...
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Conditional expected value meaning problem

I have two random variables X,Y with the same following distributions : $$P(X=0)=P(Y=0)=1-p, P(X=1)=P(Y=1)=p$$ I want to calculate $E(X|1_{\{X+Y=0\}})$. So Let's define random variable $Z=1_{\{X+Y=...
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1answer
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Equivalent Definitions of the Markov Property

Assume we have a stochastic process $\{X_n\}_\mathbb{N}$ defined on some underlying probability space that takes values in another measurable space. One of the many definitions that I have seen of ...
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25 views

Probability “on an event” vs probability “given an event”

I think I am missing something very basic. What does it mean when people say probability of something "on an event"? Is it different from conditional probability? For example, suppose I have an ...
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Why does this argument for conditional expectation fail?

I have a question and I know it is wrong. However I do not understand where I am messing up. If somebody could explain where I am going wrong, that would be great. If we have a probability space $(X,\...
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1answer
23 views

Can we take conditional expectations on a random variable that has been fixed to a value, such as $E( E\left( Y \mid X,Z=1 \right) \mid X)$?

Suppose we are trying to estimate a random variable $Y$ by conditioning on three random variables, $X,Z$, with the condition that $Z \in \{0,1\}$ is discrete. I am interested in the conditional ...
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Conditional expectation of a uniform distribution given a geometric distribution

Let N follow a geometric distribution with probability p. After the success of the experiment we define X, a uniform distribution from 1 to N. Both distributions are discrete. Find E[X|N].
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1answer
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Conditional expectation of function from $\mathbb{R}^n$ to $\mathbb{R}$

Let $f$ be a 1-Lipschitz function from $\mathbb{R}^n$ to $\mathbb{R}$, and let $\{X_i\}$ be an independent sequence of random variables, each with variance $\leq 1$. Suppose we are given a function $...
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Measurability when conditioning on a sub-sigma field

I'm currently going through a book on measure-theoretic probability. In it, we defined conditional expectation: Let X belong to $M^+(Omega, F)$ and P be a probability measure on F. For each sub-sigma-...
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96 views

Show that this integral function is a version of the expected value.

This is (part of) question 8.8 from the lecture notes on measure theoretic probability theory by P. Spreij. Let $X$ and $Y$ be random variables and assume that $(X,Y)$ admits a density $f$ w.r.t. ...
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How to approximate a Mixture distribution by a single distribution

Suppose there is a mixture of two Gaussian distributions $G = \alpha N_1 + (1-\alpha)N_2$, where $N_1,N_2$ are known Gaussian distributions, $\alpha$ is the ratio rate. I have two questions: How to ...
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1answer
35 views

Proof of an Abstract Bayes' Theorem

In Björk (2009) a Bayes' theorem is given by Assume that $X$ is a random variable on $(\Omega, \mathcal F, P)$ and let $Q$ be another probability measure on $(\Omega, \mathcal F)$ with Radon-...
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Why $\mathbb P\{X_v\in A\mid \mathcal F_s\}=\int P_{s,t}(X_s,dy)P_{t,v}(y,A)$?

Let $X$ a stochastic process and $\mathcal F_s=\sigma (X_u\mid u\leq t)$. In the book "Continuous Martingale and Brownian motion" (third edition) of Yor and Revuz, page 80 : Let $$P_{s,t}(X_s,A)=\...
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What does it mean $\mathbb E[\Phi(x,Y)]|_{x=X}$ in the book of R. Schilling (Brownian motion)?

In a book, there are the following notations that I don't understand. know conditional expectation, but here I don't get what $\mathbb E[\Phi(x,Y)]|_{x=X}$ means. Is it $\mathbb E[\Phi(X,Y)\mid X]$ ...
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The exercise from L. Breiman “ Probability ” page 76

How to solve this question ? $\textbf{15.}~$ Let $X_1 , X_2, \dots $ be independent, identically distributed random variables, $E|X_1| < \infty $, and denote $S_n = X_1 + \cdots +X_n.$ Prove ...
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42 views

Discrete random variable-expected value

For each discrete random variable $X$ and a measurable set $B$, for which $P [B]> 0$, show that $E [X | B] = \frac{E [1_{B}X]} {P [B]}$. I have $E [X | B] = \frac{\sum x_{i} * P(\{X=x_{i}\} \cap ...
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Convergence in $L_1$ of conditional expectation.

Consider a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ and $(X_n)_n$, $X$ random variables on this space. Consider a sub $\sigma$-algebra $\mathcal{G} \subseteq \mathbb{F}$. Suppose $X_n \...
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21 views

How to find the optimal decoder that minimizes the probability of error?

Suppose that the signal $X$ is drawn as... $$ X = \begin{cases} 1 & \style{font-family:inherit}{\text{with probability 1/2}} \\ 0 & \style{font-family:inherit}{\text{with probability 1/2}} \...
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If $Y_n=E(X|\mathcal{F}_n)$ for some $\mathcal{F}_n \subset \mathcal{F}$ and if $Y_n \to Y$ with probability one. Show $Y_n \to Y$ in $L^1$.

If $Y_n=E(X|\mathcal{F}_n)$ for some $\mathcal{F}_n \subset \mathcal{F}$ and if $Y_n \to Y$ with probability one. Show $Y_n \to Y$ in $L^1$. I am trying to show that $Y_n$ is uniformly integrable (UI)...
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37 views

Application of Lévy Zero-One Law

Question Let $(X_n)$ be a sequence of random variables taking values in $[0, \infty)$. Let $D=\{X_n=0\; \text{for some $n\geq 1$}\}$ and assume that $$ P(D\mid X_{1}, \dotsc, X_n)\geq \delta(x)>...
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Given $\mathbb E(5X+2)=12$ and $\mathbb E(X|Y)=Y^3$ compute $\mathbb E(Y^3)$

Given $\mathbb E(5X+2)=12$ and $\mathbb E(X|Y)=Y^3$ compute $\mathbb E(Y^3)$. I've been trying this a million different ways and can't seem to reach a final answer, I would love any suggestions!
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17 views

Random Variable convergence in $L^{q}$ space, then the conditional expectation also converges in $L^{q}$

Let $q\geq 1$, and $X_{n}, X\in L^{q}$ are random variables in probability space $(\Omega,\mathcal{F},\mathbb{P})$ and $X_{n}\rightarrow X$ in $L^{q}$, then for every sub $\sigma-$algebra $\mathcal{G} ...
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14 views

Proof Novikov theorem via Ito's lemma

Let $f(s)$ be deterministic and define $$Z(t)=\exp\Big(\int^{t}_{0}f(s)dB(s)-\frac{1}{2}\int^{t}_{0}f^{2}(s)ds\Big)$$ with $$\mathbb{E}\Big[\exp\Big(\frac{1}{2}\int^{t}_{0}f^{2}(s)ds\Big)\Big]<\...
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1answer
38 views

Find the conditional distribution of $X_t$ given $X_t+Y_t=n $ , and find $\Bbb{E}[X_t+Y_t|X_{2t}]$.

Let $\{X_t,t\ge0\}$,$\{Y_t,t\ge0\}$ be independent Poisson process with rates $\lambda$ and $2\lambda $ respectively. Find the conditional distribution of $X_t$ given $X_t+Y_t=n $ , and find $\Bbb{E}[...
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39 views

Proof that conditional density is borel measurable

I want to prove that the conditional density $f_{X|Y}: \mathbb {R^2} \to \mathbb{R}$ defined by $f_{X|Y}(x|y)= \begin{cases} \frac{f_{X,Y}(x,y)}{f_{Y}(y)} & \text{ if} f_{Y}(y)>0 \\ f_{X}(x) ...