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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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24 views

Definition of conditional mean of $ E[g(Y)X|Y=y]$ where Y is discrete.

I got the following question: $X$ and $Y$ are random variables upon the probability space $(S,F,P)$. $Y$ is discrete with range $W$. Let $g: \Bbb R\to \Bbb R$ be a function so that $g(Y)$ is a ...
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0answers
9 views

Transfering order to conditional expectation

Suppose that $Q\sim P$, $X,Y\in L^2(\mathcal{F})$ are such that $$ \mathbb{E}_{Q}[(X-Y)^2] \leq \mathbb{E}_{P}[(X-Y)^2]. $$ Under what circumstances, can we conclude that $$ \mathbb{E}_{Q}[( \...
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18 views

Expectation of normally distributed r.v conditioned on vector subspace

Let $X$ be a random variable with normal distribution $N(\mu, \Sigma), \mu \in \mathbb{R}^n$ and $V \subset \mathbb{R}^n$ be a vector subspace. I want to calculate $\mathrm{E}[X|X \in V]$ and my ...
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2answers
44 views

Using Adam's Law (Law of Total Expectation) to find expectation of residual

This may seem like a rather simple question, but I haven't been able to come up with an explanation myself or find one on the Internet. I've learned that Adam's Law states that $$E(E(Y|X)) = E(Y)$$ ...
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0answers
21 views

Formalizing conditional expectation

I need help to translate into a conditional exepectation the following problem: We have an interval of $\mathbb{R}$ of size M (say $\mathcal{M} = [0, M]$). For each element $x \in \mathcal{M}$ I ...
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0answers
25 views

Conditional expected value of order statistic

Let $\theta_1,\dots,\theta_N$ be a collection of independent RVs, ditributed uniformly on $[0,1]$. Further let $\theta^{(r)}$ be the $r$th order statistic where $\theta^{(1)}\leq\dots \theta^{(r)}\leq\...
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1answer
24 views

Prove that $\mathbf{E}(Y|\sigma(X))=\mathbf{E}(Y|\sigma(X,Z))$

Let $Z$ be a random variable independent of $(X,Y)$. Prove that $\mathbf{E}(Y|\sigma(X))=\mathbf{E}(Y|\sigma(X,Z))$ My attempt: It is obvious that $\int_A\mathbf{E}(Y|\sigma(X,Z))d\mathbf{P}=\int_A\...
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2answers
25 views

Effect of a constant on conditional expectation [closed]

Is it true that $\mathbb E[Y \vert X]$ = $\mathbb E[Y\vert X+3]$?
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1answer
22 views

Expected value of a prize that has a geometric distribution dependent on independent draws from a uniform distribution.

A number $X$ and a sequence of numbers $\{Y_n\vert n\in \mathbb N\}$ are i.i.d draws from the uniform distribution on $[0,1]$. Let $N = \inf\{n \in \mathbb N\vert Y_n > X\}$. The player conducting ...
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1answer
23 views

If $E[Y \mid X]=X$ and $E[Y^2 \mid X]=X^2$ Then $Y=X$ a.s

If $E[Y \mid X]=X$ and $E[Y^2 \mid X]=X^2$. Then $Y=X$ a.s My attempt: Using the definition of conditional expectation we have that for all $A \in \sigma(X)$ $E[Y^2-X^2 1_A]=0$ so in particular $E[Y^...
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1answer
24 views

Why is $E[f(X,Y) \mid X]=g(X)$?

Let $X,Y$ be independent and let $f$ be Borel such that $f(X,Y) \in L^1(\Omega,\mathcal{A},P)$. Moreover let $$ g(x)=E[f(x,Y)] \text{ if } \vert E[f(x,Y)]\vert <\infty \text{ and } 0 \text{ ...
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2answers
28 views

Conditional mean and variance of $X$ given that $Y=6$ for $X$ normal $N(0,1)$ and $Y$ conditionally on $X=x$ normal $N(x,1)$

Assume that $X\sim N(0,1)$ and that, given $X=x$, the conditional distribution of $Y\mid X=x$ is $N(x,1)$. Find the conditional mean and variance of $X\mid Y=6$. I proceeded as follows: $$f_{X|Y}(X|...
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0answers
11 views

Conditional Moment Generating Function of a Negative Binomial

Suppose X conditioned on K follows a Negative Binomial Distribution i.e. $X|K = k_i \sim NB(r-k_i, q)$, where $r$ is a constant and $K \sim Bin(m, q)$. I'm trying to calculate the MGF of X. So far, I ...
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1answer
34 views

Expectation of $E[Xg(X^2)]$ and conditional expectation of $E[X|X^2 ]$ [closed]

I have the next excercice: Let $X$ a random variable such that $E|X|<\infty$, the law of $X$ admits a density $f$, and the law of $X$ is symmetrical ($X = -X$ in law). For $g$ a bounded Borelian ...
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1answer
16 views

Confusion about given proof of the compensated Poisson process being a Martingale?

Given the following proof of the compensated Poisson process being a Martingale Why does the proof start with $E[N(t)-\lambda t|N(s)]$ when the question asks to prove that $X(t)$ is a Martingale? ...
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0answers
15 views

Conditional expectation of martingale and two bounded stopping times

I am trying to prove the following: Let $(X_n)$ be a martingale with respect to $(\mathcal{F}_n)$ and suppose $\tau_1$ and $\tau_2$ are bounded stopping times such that $\tau_1\le \tau_2 < B$, ...
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1answer
18 views

Conditional Exspectation of Conditional Exspectation

Picture of the Task here I got this task to solve and i am very dissapointet of myself that i can't solve this. I will write only m instead of Municipality and F instead of FederlState. For a) i got ...
3
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1answer
52 views

Compute $\mathbb{E} \left(\min(X, Y) | \max(X, Y) \right)$ for $(X,Y)$ i.i.d. uniform on $(0,1)$

Let $X, Y$ be independent random variables with uniform distribution on the interval $[0, 1]$. My task is to find $$\mathbb{E} \big(\min(X, Y) | \max(X, Y) \big).$$ I think it can be done in the ...
2
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1answer
39 views

How to solve E($2X $| Y)? Where $f(x,y) = 4e^{-2y}$

Help, please! How to solve: $E(2X| Y) = ?$ $$f(x,y) = 4e^{-2y}\;,0 < x<y \mbox{ and }\; y>0.$$ After integrating $f(x,y)$ over domain of $y$ we get marginal density of $X$: $$f_X(x) = ...
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0answers
37 views

Understanding formulas for $E(X∣X < c)$

I have $E(X∣X < c)$ where $X$ is a continuous random variable and $c$ is a given positive real value. According to this question this is equivalent to: $E[X|X<c]=\int_{-\inf}^{c}1-\frac{F(x)}{...
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1answer
30 views

Conditional expectation of a random variable conditioned to another conditionally independent variable

I have a two variables $X$ and $Y$ conditionally independent given a thirs variable $Z$. Now, assume that $Z$ can take values in ${1,...,k}$. I will have: $$ E[XY] = \sum_{i=1}^k P[Z = i]E[X|Z = i]E[...
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1answer
37 views

Expectation and variance of number of movie tickets

Let $N\sim\mathrm{Pois}(\lambda_1)$ be the number of movies that will be released next year. Suppose that for each movie the number of tickets sold is $T\sim\mathrm{Pois}(\lambda_2)$, independently. ...
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1answer
40 views

Expected value in recursive random variables

Working through some probability problems from Introduction to Probability, Blitzstein: Kelly makes a series of n bets, each of which she has probability p of winning, independently. Initially, ...
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0answers
14 views

Equivalence of independence and a conditional expectation equality

Let $E_1$, $E_2$ be Polish, let $D_{E_2}([0,\infty])$ be the space of cadlag functions with values in $E_2$ and set $\mathcal{F}_t^Y$ to be the $\sigma$-algebra generated by $Y_s$, $s \leq t$, ...
2
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1answer
32 views

Finding a mean of biased sample

An RV $X$ follows a uniform distribution over $[0,1]$. Suppose that we cannot observe the realization $x$ of $X$. Instead, we can observe a value $y$, meaning that $x\in\{y,\frac{1}{2}+\frac{y}{2},...
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1answer
48 views

Show that this inequality involving expectations holds

Let there be two events (which are disjoint and a partition of the sample space) $G$ and $B$ where $p = Pr(G)$ and $1-p = Pr(B)$. Let $X$ be a random variable and $K$ be a positive constant. Let $D = \...
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2answers
50 views

Show that $E(X|\mathcal{F}_\tau)=\sum\limits_{n\in\mathbb{N}}E(X|\mathcal{F}_n)\mathbf{1}_{\{\tau=n\}}$

If $\mathbf{E}X<\infty$ and $\tau$ is a stopping time, then $$\mathbf{E}(X|\mathcal{F}_\tau)=\sum_{n\in\mathbb{N}}\mathbf{E}(X|\mathcal{F}_n)\mathbf{1}_{\{\tau=n\}}.$$ My attempt: First assume ...
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1answer
20 views

$E(X^2|X-Y) E(X^3|X-2Y)$ for Gaussians?

For independent gaussians with following the normal distribution with expectation zero and variance one, how do I compute: $E(X^2|X-2Y), E(X^3|X-2Y)$ I know that $X-2Y$,$X+2Y$ are independent. ...
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1answer
34 views

Understanding a $\sigma$-field

Problem Let's consider $n$ Bernoulli trials with the probability of getting a success equal to $p$. My task is to find the expected value of getting a success in the first trial on condition that we ...
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1answer
18 views

Apparent inconsistency in the expectation of max of IID RVs

$X_1, X_2, \dots, X_n$ are $n$ IID RVs. $Z = \mathrm{max}(X_1, X_2, \dots, X_n)$. Now define $I$ as follows: $I = i$ when $X_i$ is the maximum of $X_1, X_2, \dots, X_n$. Using law of total ...
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1answer
37 views

A problem on equivalent definitions of Markov property

Suppose that $X, (\Omega,\mathcal{F}),\{P^x\}_{x \in \mathbb{R}^d}$ is a Markov family with shift operators $\{\theta_s\}_{s \ge 0}$ and for every $x \in \mathbb{R}^d,s \ge 0, G \in \mathcal{F}_s$ ...
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0answers
31 views

Expected ratio of turns for complicated ball drawing game

I am stumped once again by another expectation value question. The question is as follows. Alice, Bob and Charles play a game. In front of the players are four urns, each containing an equal number ...
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1answer
9 views

Confidence Interval of number of red marbles among 100 marbles where proportion of red marbles is uniformly distributed

A bag contains 100 marbles of colors red and black. The proportion of red color marbles is uniformly distributed between 0 and 1. How do I compute the confidence interval of the number of red marbles? ...
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17 views

Question on conditional variance in the book Introduction to Probability 2nd edition [duplicate]

I read the book of Introduction to Probability SECOND EDITION by John N. Tsitsiklis et al. and in Sec. 4.3 Conditional Expectation and Variance Revisited, I think the $var(X|Y)$ is not $nY(1-Y)$ but $...
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1answer
25 views

Proving That a Version of the Law of Total Probability Follows from Adam's Law

I have a homework question that asks: Show that the following version of LOTP follows from Adam’s law: for any event A and continuous random variable X with PDF $f_X$: $$ P(A) = \int_{- \infty}^{\...
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2answers
16 views

Does conditional expectation imply anything about the expected value of the product of two random variables?

If $E[U|X]=0$ then $[XU] = 0$ If $E[XU]=0$ then $[U|X] = 0$ Which of the two statements above are true? This is my thought process for the first one: if $E[U|X]=0$ then $E[U]=0$ and if U and X are ...
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0answers
26 views

Conditional Expectation of One Member of a Multinomial Distribution, Conditioned on a Second Member

I have a homework problem, which asks: Let X = $(X_1, X_2, X_3, X_4, X_5) \sim \text{Mult}_5(n, p)$ with p = $(p_1, p_2, p_3, p_4, p_5)$. (a) Find $E(X_1 | X_2) \text{ and } Var(X_1 | X_2)$. (b) ...
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0answers
30 views

Calculating a conditional expectation (probability theory)

Let $X$ and $Y$ be two random variables with the joint probability density function $f_{xy}(x,y)=(15/4) x y^2$ for $0 < 2 y \leq x \leq 2$ and $0$ otherwise. Calculate ${\cal E}[X^2+Y^2 \leq 1 | X ...
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2answers
41 views

How to prove that $E(Y|D=1)=E(DY)/E(D)$

How to prove that $$E(Y|D=1)=E(DY)/E(D)$$ and $$E(Y|X,D=1)=E(DY|X)/E(D|X),$$ where $D$ is a binary variable and takes value of 0 and 1.
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0answers
31 views

The expectation of a geometric random variable where its parameter is uniform

First thanks for any help editing my text. If a random variable $X$ has a geometric distribution with parameter $P$ where $P$ itself is a random variable and uniformly distributed from $0$ to $1-1/n$,...
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0answers
35 views

How to rigorously prove $E(X)=E(X|A)P(A)+E(X|A^c)P(A^c)$?

X is a random variable while A is an event. I try to prove $$E(X)=E(X|A)P(A)+E(X|A^c)P(A^c)$$ Assuming $X$ is a continuous random variable, if $$E(X|A)=\int_{x\in A}xf_{X|A}(x)dx=\int_{x\in A}xf_{X,A}...
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1answer
15 views

Use of law of total expectation without checking integrability

$\newcommand{\E}{\mathbb E}$In basic probability classes, people often use the formula, namely the law of total expectation $$\E[X]=\E[\E[X\mid Y]]$$ without checking integrability of $X$. I can't ...
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2answers
23 views

relations between expectation conditioned on subsets and conditional expectation

Let $\mathbb{E}[|Y|]< \infty$. Set $\mathbb{E}[Y |B] = \frac{\mathbb{E}[Y \mathbb{1}_B]}{P(B)}$ for $B$ with $P(B) > 0$. Suppose that $\mathbb{E}[Y | X \in [a,b]] \in [a,b]$ for all $a<b$ ...
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2answers
50 views

Conditional expectation of number of trials

Consider $n$ independent trials, each of which results in one of the outcomes $\{1, ..., k\}$, with respective probabilities $p_1, p_2, ...,p_k$ where those probabilites sum to $1$. Let $N_i$ denote ...
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1answer
21 views

Example of E(E(X|F)|G) \neq E(E(X|G)|F) [duplicate]

Can you find an example where E(E(X|F)|G) $\neq$ E(E(X|G)|F) (F and G is $\sigma$-field in probability theory)
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0answers
63 views

Conditional expcetation of a function of multivarite normal random variables

Say we have some function of bivariate standard normal random variables $f(x_1,x_2)$ which we approximate with first 2 terms of its Taylor series, $$z = f(x_1,x_2) \approx f(0,0) + f_1(0,0)x_1 + f_2(...
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0answers
24 views

Conditional expectation, conditioning on a linear combination

If, by hypothesis, a certain random variable v is independent from another random variable w, and we define a third random variable u such that u = v - w, can one say that E(w|u) = E(w| v – w) = E(w|-...
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1answer
12 views

Finding Expectation of Continuous Joint Density

I am working on a problem and am a bit confused. The problem: Consider R.V. X and Y distributed on the triangle {(x,y) $\in$ $\mathbb R^2$| 0$\le$X$\le$1, 0$\le$Y$\le$x} p(x,y) = 2, 0$\le$X$\le$1, ...
2
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2answers
27 views

Martingale of function of RV's

Let $\phi_n:\mathbb{R}^n\to\mathbb{R}$ for all $n\in\mathbb{N}$ be a measurable function, and $X_1,...,X_m$ independendent rv's and $\Sigma_m=\sigma(X_1,...,X_m)$. Further suppose $E(\phi_{m+1}(x_1,......
1
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1answer
40 views

Conditional expectation of a function with independent random variables

If $X_1, ..., X_{n+1}$ are independent real random variables and $h:\mathbb{R}^{n+1} \to \mathbb{R}$ a Borel function. Now taking the conditional expectation $\mathbb{E}[h(X_1, \ldots, X_{n+1})| \...