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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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36 views

What is the expectation of $X$ given $X$

Hi im trying to understand conditional expectation and conditional probability based on sigma algebras. Therefore an answer in that flavour would be most useful. So in a physical sense I can see ...
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1answer
18 views

Sigma-field expectation problem

Let $\mathcal{F}$ and $\mathcal{G}$ be two σ-fields and suppose $\mathcal{F}\subset\mathcal{G}$. Let $X$ be a $\mathcal{F}$-measurable random variable. Show that $E(X^2)\ge E(E(X \mid \mathcal{G})^2)$....
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1answer
13 views

Conditional independence and conditional expectation for joint Gaussian vectors.

I am reading Hajek's book "Random Processes for Engineers". In example 4.5 it says: Let $X,Y,Z$ be joint Gaussian vectors. Then $X$ and $Z$ are conditionally independent to each other given $Y$ if and ...
4
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1answer
54 views

Why are these two definitions of Markov property equivalent?

Question Suppose that $S$ is a finite or a countable subset of $\mathbb R$ and $(\xi_n)_{n\in\mathbb N}$ is an $S$-valued sequence of random variables. Then are these two definitions of Markov ...
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1answer
16 views

How to geometrically interpret conditional expectation property tower rule??

I have a question on how to interpret conditional expectation its properties geometrically. There are two properties of conditional expectation in particular that I’m trying to interpret: Given a ...
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1answer
19 views

Relationship conditional expectation and random variable under specific constraint on its values

I am trying to establish a relationship between the following conditional expectation and random variable based on the a given identity: Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability ...
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0answers
45 views

How can I calculate this conditional expectation?

I tried to solve this problem but I can't find an easy way to do it. I have n short boards and m long boards (twice the short one). I randomly choose one board at a time to fill my floor: if I choose ...
3
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0answers
30 views

Existence of random variable with some conditions

Let $(Z_k)_{k\in\mathbb N}$ be a Markov chain on $(\Omega,\mathcal F,\mathbb P)$ taking values in $(\mathbb N,2^{\mathbb N})$. Let $p\in [0,1]$ and define $F:\mathbb N\times 2^\mathbb{N}\to [0,1]$ ...
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17 views

Understanding the expectation over training set

I don't know is this the right place to ask, but I will try... I am trying to read the Elements of Statistical Learning Tibshirani, Hastie and Friedman, however I have a problem with understanding ...
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1answer
21 views

Notation for conditional expectation using integral measure

Hi I am struggling to understand this notation for conditional expectation: (Say $X_{t}$ is a process that takes values in $\mathbb{R}$) then $$E[f(X_{t})|X_{0}=x]=\int_{\mathbb{R}}f(y)p_{t}(x,dy)$$ ...
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1answer
26 views

Finite-dimensional conditional distributions of a Markov process

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $I\subseteq\mathbb R$ $(\mathcal F_t)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\mathcal E)$ be a measurable space $X$ be ...
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23 views

Weak continuity of conditional expectation

Let $(\Omega,\mathscr{F},\mathbb{P})$ be a probability space, let $\mathscr{P}(\mathbb{P})$ be the collection of probability measures dominated by $\mathbb{P}$, and let $\mathscr{G}$ be a sub-$\sigma$-...
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1answer
31 views

Show some property of a Markov process

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $I\subseteq\mathbb R$ $(\mathcal F_t)_{t\in I}$ be a filtration on $(\Omega,\mathcal A)$ $(E,\mathcal E)$ be a measurable space $X$ be ...
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1answer
25 views

Prove a simple property of conditional expectation

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $\mathcal F$, $\mathcal G$ and $\mathcal H$ be $\sigma$-algebras on $\Omega$ with $\mathcal F\subseteq\mathcal G\subseteq\mathcal H\...
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0answers
16 views

Conditional expectation of a function of two random variables given one of them

Given two random variables $X_1,X_2$ how does one prove $$E[g_1(X_1)g_2(X_2)|X_2] = E[g_1(X_1)|X_2]g_2(X_2) $$ I can see the intuition that since $X_2$ is given, the piece depending on it should come ...
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0answers
19 views

Does conditional expectation exist if marginal expectation exists?

Given two random variables $X,Y$ on the same probability space, given that $E[Y]$ exists, does $E[Y|X=x]$ exist for all $x$ such that $f_X(x)>0$? My argument : no it does not. If $E[Y|X=x]$ is a ...
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1answer
52 views

Conditional Expectation of E(2X+Y|X-Y=1)

I met these two problems. The first question reads: what is $E(X|X+Y=1)$ given that $X$ and $Y$ are both independent standard normal random variables. The second reads that $X$ and $Y$ are ...
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1answer
14 views

Comparing $ E(Y|X) $, $ E(Y|p(X)) $, $ E(Y|X, p(X)) $

Consider 2 random variables $Y,X$ and a function $p(X)$. I would like to understand the relation between these 3 conditional expectations $$ E(Y|X) $$ $$ E(Y|p(X)) $$ $$ E(Y|X, p(X)) $$ My intuition ...
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1answer
39 views

Prove (or derive) $𝔼[X|X≤x]⩽x<𝔼[X|X>x]$ [closed]

How can I prove (or derive) the following law on conditional expectation: $$𝔼[X|X≤x]⩽x<𝔼[X|X>x]$$ It appears in this post but I don't know where this identity comes from.
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0answers
40 views

Expectation of quadratic term with 2 random variables

I'm trying to understand, why the MSE of two Random Variables $\Theta$ and X is: $ E[] $ is the expectation and $f_X(x)$ is the probability density function of X $$ E[(T(X)-\Theta)^2] = E[E[(T(X)-\...
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1answer
47 views

Is this proof for $\mathbb{E}(XY|X)=X\mathbb{E}(Y|X)$ correct and standard?

We know that $\mathbb{E}(XY|X)$ is itself a random variable with respect to $x$. So, to prove the given statement in the title, it suffices to treat $\mathbb{E}(XY|X)$ as a function of $x$ and prove ...
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0answers
20 views

Deriving matrix equation for first passage times in Markov Chains

I'm trying to understand a step in my class notes on deriving the matrix equation which allows us to compute expected first passage times for finite state Markov Chains. The notes proceed as follows, ...
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2answers
105 views

Conditional expectation of random sums

A few days ago I came across the following problem: Let $\{X_n\}_{n\ge 0}$ and $W$ be random variables. Suppose $W : \Omega \to \mathbb{N} \cup \{\infty\}$ and $S_W := \sum_{i = 0}^W X_i \in L^1$....
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0answers
36 views

Naturality of Conditional Expectation

Context Let $\mathscr{G}$ be a $\sigma$-subalgebra of $\mathscr{F}$ and let $\mathbb{P}$ be a probability measure on a polish-space $X$. Define the category $\mathfrak{C}$ of random-elements in $L^...
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0answers
15 views

Conditional expection of discrete random variable

The problem Suppose $X_1$ and $X_2$ are random variables of the discrete type which have the joint pmf $p(x_1, x_2) = (x_1 + 2x_2)/18$, $(x_1, x_2) = (1, 1), (1, 2), (2, 1), (2, 2)$, $0$ elsewhere. ...
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2answers
35 views

Formalise a trick of computing expectations

First let's start with a good old problem: Consider flipping a fair coin. Let $N_1$ be the number of flips you need to see the first heads (including the last flip of heads). Compute $\Bbb E(N_1)$. ...
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0answers
17 views

Law of total expectation with a conditional event

I know that, given that $\{A_i\}$ is a finite or countable partition of the sample space, then, for any event $X$, we have: $$\mathbb{E}[X] = \sum_i \mathbb{E}[X | A_i]p(A_i).$$ Consider now ...
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2answers
38 views

Expected number of heads when flipping until both heads and tails are encountered

Consider a coin that lands heads with probability $p$. If I flip the coin until both heads and tails are encountered, let $X$ be a random variable representing the number of heads flipped. What is $\...
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2answers
43 views

Intuition behind the expectation of a conditional expectation

Let $(\Omega, \cal F, P)$ be a probability space and $X,Y$ two random variables defined on $(\Omega, \cal F, P)$. I stumbled upon the equality $$E[E(X \vert Y)]=E(X)$$ Observing that $$\begin{align}...
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1answer
8 views

Expectation of Squared Condtitional expectation and the tower property

What can I say about $E(X \hat{X})$ where $\hat{X}$ is a version of $E(X|\mathcal{G})$, where $X \in \mathcal{L}^2(\Omega,\mathcal{F},\mathbb{P})$ and $\hat{X} \in \mathcal{L}^2(\Omega,\mathcal{G},\...
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1answer
30 views

Explain conditional expectations and taking out measurable random variables

We have the following property of conditional expectations from the point of veiw of measure theory ($X$ and $Y$ are two random variables): $$\mathbb{E}(XY\mid\mathcal{G})=X\operatorname{\mathbb{E}}(...
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0answers
63 views

What is wrong with below proof that E[XY|X] not equal to E[Y|X]

The equality above is wrong. But what was the issue with below derivation? Set $T=XY$ $$ E[YX|X=x]=\int_a^b t f_{T|X=x}(t) dt $$ perform change of variables, $dt=x dy$. So $$ E[YX|X=x]=\int_{\frac{a}{...
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1answer
24 views

Conditional expectation of product of martingales

I did not find the following question addressed nor a counterexample or proof. For a continious martingale does, $\mathbb{E}(X_{t}X_{s} \mid \mathcal{F}_{u})=X_{u}\mathbb{E}(X_{s} \mid \mathcal{F}_{...
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1answer
66 views

Conditional Expectation of $X$ given $aX+Y$ where $a$ is constant

If $X$ and $Y$ are independent $U~(-1,1)$ distributed random varibles. What would be $E(X|Z=aX+Y)$ where $a$ is a constant? I start to solve this by $$f(x|Z)=\frac{f(Z|x)\cdot f(x)}{f(Z)}$$ where $...
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0answers
16 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 ...
2
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1answer
54 views

If $E(Y\mid X) = E(Y)$, do we have $X,Y$ independent? [closed]

We know that if $X,Y$ are independent, then $E(Y\mid X) = E(Y)$. But what about the converse?
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1answer
28 views

Normal distribution conditional on a sub-affine space

This is related to this problem. Now let's for the time being keep aside the unorthodox "probabilistic" statements and focus on the canonical deterministic part only. Suppose $w\sim N(\mu, \Sigma)$ is ...
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1answer
63 views

Seating arrangements: Question about the book solution and summation indices

$20$ people are to be seated at seven tables, three of which have 4 seats and four of which have 2 seats. If the people are randomly seated, find the expected value of the number of married couples ...
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0answers
24 views

Normal distribution conditional on both deterministic and probabilistic statements

Let $w \sim N(\mu,\Sigma)$ be multivariate normal and let $w_1,w_2$ be two non-overlapping sub-vectors of $w$ respectively. Then how would $w$ be distributed conditional on the following statements $$...
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0answers
17 views

Show that E(Y | X) = E(E(Y | X, W) | X). The right hand side may sometimes also be written as E(E(Y | X, W) | X) = E(E(Y | W) | X).

We know that, E(E(X|Y)) = E(v(Y)) = E(E(X|Y=y)) = E(X), where v(Y) = E(X|Y=y). Hence, E(E(X|Y)) = E(X). Note: Let X and Y be discrete or jointly continuous random variables. The conditional ...
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1answer
44 views

Estimation of $\operatorname{var}[P(X > 20\mid p)]$ while $X\mid p\sim \text{Bin}(100, p)$ and $p \sim \text{Beta}(60, 30)$

I have $p \sim \text{Beta}(60, 30)$. And I have to estimate $\operatorname{var}[P(X > 20) | p]$ while $X \sim \text{Binomial}(100, p)$. I think I can sample $p$, calculate $P(X > 20)$ by ...
2
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1answer
70 views

Coupon Collecting Problem: Condition on the order of coupons you see

There are four types of coupons (call them types $1,2,3,4$) with $p_1 = p_2 = 1/8$ and $p_3 = p_4 = 3/8$. You collect coupons until you have a full set. Q1: What is the probability that after ...
14
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5answers
2k views

Four coins with reflip problem?

I came across the following problem today. Flip four coins. For every head, you get $\$1$. You may reflip one coin after the four flips. Calculate the expected returns. I know that the expected ...
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2answers
62 views

Coupon Collecting Problem: $4$ coupons with $p_1 = p_2 = \frac{1}{8}$ and $p_3 = p_4 = \frac{3}{8}$

This is from Ross. I know how to solve everything but (d). The book answer is $\frac{123}{35}$. There are $4$ different types of coupons, the first $2$ of which compose one group and the second $2$...
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2answers
31 views

let $X$ be a random variable , $A\subset B$ is that $\mathbb{E} [X|A] \leq \mathbb{E}[X|B]$ true in general?

1.let $X$ be a random variable , $A\subset B$ is that $\mathbb{E} [X|A] \leq \mathbb{E}[X|B]$ true in general ? 2 what about If $X$ is Gaussian random variable , $A\subset B$ is that $\mathbb{E} [...
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0answers
28 views

Conditional expectation with 2 events

Using the definition of conditional expectation given an event I need to show $$ E (I_B | A) = P(B|A).$$ So far I have \begin{align*} E (I_B | A) &= \sum_x 1\cdot P(I_B = 1 |A) P(A) + 0 \cdot P(...
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0answers
20 views

Variance of stochastic variable with X and Y

if $X$ and $Y$ are dependent stochastic variables where $Var(x) = 1.95$, $Var(Y)=0.9$ and $Cov(X,Y)=0.8$. Find the variance of $Z = -4X+4Y-6$ I tried using $Var(X+Y) = Var(X) + Var(Y) + 2cov(X,Y)$ ...
1
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1answer
20 views

Branching process expectancy given an initial condition

Given a branching process {$X_n$} with offspring probabilities $p_0 = 1/6, p_1 = 1/3,$ and $p_2 = 1/2$, find ${E[X_2|X_0 = 10]}$. I know that $E[X_n|X_{n-1}] = X_{n-1}\mu$. I tried stating that $E[X_2|...
0
votes
1answer
40 views

Conditional expectation value of battery drawing problem

A box contains $a$ batteries of which $d$ are dead. The batteries are tested randomly, one by one. Every time that a good battery is drawn, it is returned to the box. When a dead battery is drawn, it ...
1
vote
2answers
32 views

Word probability problem about conditional expectation

A spam filter checks each incoming message to be classified as either spam or non-spam, and only messages classified as non-spam are to be delivered to each inbox. On average, however, one percent of ...