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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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Calculate $\mathbb E [\mathbb E[ \sum_{n=0}^{\infty}\mathbb E [T | X_T = n]\mathbb P[X_T=n] | T]]$, where $T\sim$ Exp and $(X_t)$ a Poisson process

Assume $T \sim \text{Exp}(\mu)$ and let $X = (X_t)_{t \geq 0}$ be a Poisson process of rate $\lambda$. I am trying to calculate $\mathbb E [\mathbb E[ \sum_{n=0}^{\infty}\mathbb E [T | X_T = n]\mathbb ...
hm1912's user avatar
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Conditional expectations preserve convergence in measure

Suppose that $(X_n)_{n\in\mathbb{N}}$ is a sequence of random variables defined on a probability space $(\Omega, \mathcal{F}, P)$ and $\mathcal{G}\subseteq\mathcal{F}$ be a sub-$\sigma-$algebra. Then ...
naveenraj03's user avatar
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Confused by conditional expectation calculation

Let $X$ be a random variable following a triangular distribution with pdf: $$f(X = x) = 2x \text{,}\quad 0 \leq x \leq 1.$$ $$\hspace{0.8cm}\qquad= 0 \text{,}\quad \text{otherwise}.$$ For a given ...
gaussplustwo's user avatar
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1 answer
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Normal distribution and conditional expectation

Why is $E(X|X>a) \left( 1- \Phi\left( \frac{ln(a)-\mu}{\sigma} \right) \right)=E(X) \left( 1-\Phi\left( \frac{ln(a)-\mu}{\sigma} -\sigma \right)\right)$? Maybe it's easy but I do not see it.
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How to prove this formula about expectation ?

The continuous-time Markov chain has an infinitesimal generating element Q. For all $f \gt$ $0$,define $$Z(t)=f(X_t)\exp\left(-\int_0^t\left(\frac{Qf}{f}\right)(X_s)ds\right).$$ Define $\tau_n$ as the ...
Jie's user avatar
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Applying the law of total variance [closed]

Say we have a sample of 100 normally distributed payments, with mean=1000 dollars and standard deviation= 100 dollars. 10% of these payments were made in error and should be refunded their full ...
duke's user avatar
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If $T$ is sufficient, then there's a probability $\mathbb{P}_0$ such [...] $\mathbb{E}_{\mathbb{P}_0}(X|T)=E_P(X|T)$ a.e. for all $P\in\mathfrak{M}$

Let $(\Omega,\Sigma)$ be a measurable space and $\mathfrak{M}$ a family of probability measures. Suppose that exists a $\sigma$-finite measure $\mu:\Sigma \to\overline{\mathbb{R}}$ such $P\ll \mu$ for ...
rfloc's user avatar
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Can we conclude that $X$ is $\mathcal{G}$-mensurable if $X=\mathbb{E}[X|\mathcal{G}]$ a.e.?

Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, $\mathcal{G\subseteq F}$ be a sub-$\sigma$-algebra and $X:\Omega\to \mathbb{R}$ be an integrable random variable. Can we conclude that $X$ ...
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If x is uniform, what is 𝐸[𝑥∣𝑥>𝑥′] for 𝑥′∈[0,1]?

This is a very simple question but I am failing to check if my answer is correct: Given that $x \sim U[0,1]$, what is the $E[x |x > x']$ where $x'$ is within $[0,1]$? My answer would be that its $(...
learner's user avatar
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$\mathbb{E}[Y \mid X_1] = 1 + (1 - X_1)\mathbb{E}[Y]$ [closed]

Let $X_1,X_2...$ be iid and assume: $P(X_1 = 1) = 1 - P(X_1 = 0) = p \in (0, 1)$ Let Y be defined as: $Y = \inf\{n \in \mathbb{N} : X_n = 1\}$ So that $Y=n$, where $n \in \mathbb{N}$ if and only if $...
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Monotone likelihood ratio property and monotone hazard rates

Suppose $f(x)$ and $g(x)$ are pdfs with positive support on $[0, \infty)$ and have the monotone likelihood ratio property. Then, they have monotone hazard rates: $$\frac{f(x)}{1-F(x)} \leq \frac{g(x)}{...
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2 votes
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Exercise 7.14 First course in probability

Exercise 7.14 First course in probability : An urn has $m$ black balls. At each stage, a black ball is removed and a new ball that is black with probability $p$ and white with probability $1 - p$ is ...
MathematicsBeginner's user avatar
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1 answer
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$E[X|X^2+Y^2] = 0$ when $X$ and $Y$ are independent standard normals.

I need to show that $E[X|X^2+Y^2] = 0$ when $X$ and $Y$ are independent standard normals. $E[X|X^2+Y^2] = \int \frac{f(x, x^2+y^2)}{f(x^2+y^2)} \frac{1}{\sqrt{2\pi}}e^{\frac{x^2}{-2}}dx$. We want to ...
Adam's user avatar
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Finding the expectation of Gaussian a RV conditioned on affine transformation of jointly Gaussian Vector

Trying to solve a problem: Consider the jointly gaussian vector $[X_0, X_1]^T$ where $X_0$ and $X_1$ are independent, zero-mean Gaussian RVs with variance $\sigma_0^2, \sigma_1^2$ respectively. Then ...
stochastics1q2490230958's user avatar
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2 answers
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Inequality regarding expectations

I have a simple question that somehow eludes me. Namely: is it always true that for an integrable r.v. $X$ and some sigma-algebra $\mathcal{H} \subset \mathcal{F}$, where $\mathcal{F}$ is from a ...
aarin.kailon's user avatar
2 votes
1 answer
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Alternative Proof of Hoeffding's Lemma / solve the equation $E[1_A | X] = (1+X)/2$ given $X$

In the context of proving the Hoeffding Lemma I came across a slightly weaker statement in the form of an exercise: "If $X$ is a real valued random variable and $|X| \leq 1$ a.s. then there ...
2000mg Haigo 's user avatar
1 vote
2 answers
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Definition of Left-Closable Martingale

I am currently studying martingales with Resnick's book A Probability Path. He defines a martingale as closed on the right if there is an $X \in L_1$ such that $X_n = \mathbb{E}[X \mid \mathcal{B}_n]$ ...
picklechu's user avatar
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conditional expectation of non-negative variable

In Angrist's book Mostly Harmless Econometrics, section 3.4.2, it says $$ E[Y_i|D_i]=E[Y_i|Y_i>0,D_i]P[Y_i>0|D_i] $$, where $Y_i$ is a non-negative variable ($Y_i$ can be $0$) and $D_i$ is a ...
Kozack51's user avatar
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How to define conditional distribution in terms of expectation value

Let $(\Omega, \mathcal{A},\mathbb{P})$ be a probability space and $Y:(\Omega, \mathcal{A})\rightarrow (\Omega', \mathcal{A}')$ be a random variable. So usually conditional probability is defined via ...
guest1's user avatar
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1 answer
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A and B flip a fair dice until 6 occurs. Why was my calculation wrong?

A and B flip a fair dice until 6 occurs. I have read this post and tried to work out the expectation of flipping on A wins. But it seems something goes wrong. Following is my work: A wins iff $6$ ...
Sven2009's user avatar
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2 answers
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Is it true that $E[X|Y] = \rho \frac{\sigma_X}{\sigma_Y} Y$ if $X$ and $Y$ are not jointly gaussian?

Let $X$ and $Y$ be centered normal (Gaussian) random variables. Let $\rho:= \frac{E[XY]}{\sigma_X \sigma_Y}$ where $\sigma_X^2 = E[X^2]$ and $\sigma_Y^2 = E[Y^2]$. It is known that if $(X,Y)$ is a ...
ProbabilityLearner's user avatar
3 votes
1 answer
46 views

Expectation of the indicator function

Define: For $n \geq 0$, on note $X_n=(n+1) \mathbb{1}_{[n+1,+\infty}$, and $\mathcal{F}_n=\sigma(\{1\},\{2\}, \ldots,\{n\},[n+$ $1,+\infty[)$ and $\forall k \in \mathbb{N}^*, \mathbb{P}(\{k\})=\frac{1}...
phi's user avatar
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1 answer
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Expectation of the squared conditional expectation

I am considering the expectation of the squared expectation, as asked here but with no answer and so wanted to get the communities thoughts. Since $E[Y|X]$ is not independent with itself then the ...
InvestingScientist's user avatar
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17 views

Conditional Expectation Notation in ARCH Model

I'm new to ARCH models, and I have a question about the correct notation for expressing the conditional expectation of the return at time $t(r_t)$ given the information available up to time t-1. I'd ...
Newbie's user avatar
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2 votes
3 answers
160 views

Expected value of $X_N$ with smallest index $N$ for which $\sum_{i=1}^N X_i$ exceeds $1$ when $X_i$ are uniformly distributed

From an interview book, where the answer is not so clear I believe. You keep generating $\mathcal U_{[0,1]}$ iid random variables until their sum exceeds 1, then compute the expected value of the last ...
marco's user avatar
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1 answer
34 views

joint distribution and conditional expectation

I am confused about conditional expectations. Let $(\Omega, \mathcal{F},P)$ be a probability space. Next, let $X$ and $Y$ be random variables on this space. Next, let $E[X|\sigma(Y)]$ be conditional ...
AnTlr's user avatar
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0 answers
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Independence between sums of $Z_k$

Let's assume you are faced with the following random variables $$ X_i = \sum_{k}\alpha_{i,k}Z_k + \epsilon_i$$ where $Z_k,\epsilon_i$ are i.i.d. standard normal random variables and $\alpha_{i,k} >...
BMBE's user avatar
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Iterated Conditional Expectation Problem Related to Random Walks with Retirement

I am currently working through the results presented in the paper titled "Consistent Price Systems and Face-lifting Pricing under Transaction Costs" and authored by P. Guasoni, M. Rásonyi ...
ShaftSinker's user avatar
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24 views

Law of iterated expectation application $E\left\{ \frac{1}{E \left(Y|X \right)} \right\}$

Let $X$ and $Y$ be two continuous positive random varianbles, that are not independent. In addition, all of the stated expectations are bounded away from zero. We know that $ E(X)^{-1} \neq E(1/X)$ ...
A_Mondial's user avatar
1 vote
1 answer
48 views

Intuitively, why does the conditional expectation of X given the trivial sigma algebra equals the expected value of X itself?

If we are conditioning X given the trivial sigma algebra then we get the expectation of X, its proof is trivial but intuitively what does this case represent ?
Pat's user avatar
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Marginalizing the product of two conditional distributions sharing a joint

I have a joint normal distribution $$\mathcal{N}\left(\left.\begin{bmatrix}X\\ Y\\ Z\end{bmatrix}\right|\mu, \Sigma\right) $$ I want to find the following quantity $$\int \mathcal{N}(X|Y)\mathcal{N}(Z|...
cdmath's user avatar
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1 answer
20 views

Understanding conditional excpected value as function

I have a problem with understanding what is the difference between: $\psi (y) := \int_{\mathbb{R}^m} x \cdot f_{X|Y}(x | y) \ dx$ ,therefore $\psi (Y)$ = $E(X \mid Y)$ and $\phi (y) := \int_{\mathbb{...
Davidoff's user avatar
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1 answer
36 views

Conditional probability involving two random times, where only the distribution of one of them is used.

Consider two random times $\tau_{1}$ and $\tau_{2}$ defined on a common probability space $\left(\Omega, \mathcal{G}, \mathbb{Q}\right)$ with $\mathbb{Q}(\tau_{k}=0)=0$ and $\mathbb{Q}(\tau_{k}>t)&...
CA-Math's user avatar
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1 answer
60 views

derivate discounted payoff

Let $S(t)$ be the stock price at time $t\geq 0$ with $S(0)=s_0$ and let $\Pi(S_t)=\max\{K-S_t,0\}=(K-S_t)_+$ the payoff of an american put with strike price $K$. How can I calculate the derivate $\...
Robert's user avatar
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How to ensure that integrals are remain within limits of domain of the variables?

I have a Manager who has no information about the profit from investment into two bonds, other than the fact that they are independently drawn from uniform distribution ${m, 1}$ where, m>0. Profit ...
Elina Gilbert's user avatar
1 vote
1 answer
61 views

What's the relationship between $(X_i)$ and $(X_i - \mathbb{E}[X_i | X_{<i}])$?

Let $X_1,\cdots,X_n$ be $n$ random variables on the same probability space $(\Omega, F, P)$, all with expectation $0$. Define $Y_i=X_i - \mathbb{E}[X_i | X_1, \cdots, X_{i-1}]$. Is it true that for ...
ryanstar's user avatar
0 votes
1 answer
70 views

Difficulty understanding the conditional expectation

I'm currently having a confusion dealing with the conditional expectation. Let's recall the definition first: Let $X$ be an integrable function (or a random variable) defined on a probability space $(\...
MintChocolate's user avatar
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0 answers
91 views

Law of total variance on function of random variables

Let's take the following random variables $X,Y,Z_1,...,Z_K$, on the same probability space (or associated with the same experiment), where we know that $$ Y = \sum_{k=1}^K\beta_kZ_k$$ Let's say I am ...
BMBE's user avatar
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1 answer
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Tower law and iterated expectation

Let's take 3 random variables $X,Y,Z$ on the same probability space (or associated with the same experiment), then applying the tower law in an iterative fashion we get: $$ E[X] = E_Y[E_{X|Y}[X|Y]] = ...
BMBE's user avatar
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3 votes
2 answers
129 views

Need to compute the conditional expectation

Note: I have edited the question to add more context to it. Please provide me with feedback. One unbiased die is thrown 10 times. For $1 \leq i \leq 6$, let $X_i$ denote the number of times $i$ ...
Kanishk Alok Banthia's user avatar
1 vote
1 answer
92 views

Law of total variance on conditional expectation

Let's take 3 random variables $X,Y,Z$ on the same probability space (or associated with the same experiment), then the law of total variance states that: $$ V[X] = V[E[X|Z]] + E[V[X|Z]] $$ Then what ...
BMBE's user avatar
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1 answer
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A question on conditional expectation of a random variable

Consider the joint probability density function: $$f(x_1,x_2)= \begin{cases} 2e^{-2x_1}, & \text{ for } 0 \le x_2 \le x_1 < \infty \\ 0, & \text{ elsewhere} \\ \end{cases} $$ Find the ...
MathRookie2204's user avatar
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2 answers
80 views

Joint distribution of two conditional distributions

I am trying to understand how a joint distribution is formed when two regular conditional distributions are involved that are conditional with respect to different random variables. Let $(\Omega, \...
guest1's user avatar
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0 answers
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How to calculate $E(X|X\ge Y)$ if X is dependent with Y

Now we have several independent and identically distributed random variables following the uniform distribution on the interval [0, 1].They are denoted as $x_1, x_2, x_3, ..., x_m$ and $y_1, y_2, ..., ...
Randy's user avatar
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1 answer
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On the proof of convergence of Pòlya urns

I'm reading the proof of Proposition $2$ of the Appendix here. The proposition states Let $d\ge 2$ and $S\ge 1$ be integers. Let also $(\alpha_1,\dots,\alpha_d)\in\mathbb{N}^d \setminus\{ 0 \}$. Let $...
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Intuition why pulling out $\mathcal{G}$-measurable $X$ of $E[f(X,Y)|\mathcal{G}]$ requires $Y$ indep of $\mathcal{G}$

Given $\mathcal{G}$ a sigma-field and $X$ a $\mathcal{G}$-measurable random variable, on an intuitive level, why do we need $Y$ to be independent of $\mathcal{G}$ to pull out $X$ from the conditional ...
edamondo's user avatar
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1 vote
1 answer
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Why do we need square integrability in showing $E(Y|X)$ minimizes expected quadratic loss?

I've read about that $E(Y|X)=\underset{f(x)\in \mathcal{F}}{\arg\min} E(Y-f(X))^2$, where $\mathcal{F}$ is the set of all square integrable functions in $x$. The proof of this result is simple and ...
ExcitedSnail's user avatar
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1 answer
27 views

Bayesian updating on expectation with many candidates

I have an individual selected for a job. He was competing against 10 candidates. All individuals are independently drawn from uniform distribution U(0,1). Him being chosen means he is better than the ...
Elina Gilbert's user avatar
0 votes
1 answer
40 views

Limit of sequence of continuous time martingales is a martingale

Let $X^1, X^2, X^3,\ldots $ be $L^2$-integrable martingales. Assume that for each $t\in[0,T]$ there exists an $X_t\in L^2(\Omega)$, such that $X_t^n\rightarrow X_t$ in $L^2(\Omega)$ for $n\rightarrow\...
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1 vote
1 answer
50 views

$E[\exp(-sX)\exp(-sY)]$ for two identically distributed, but correlated random variables X and Y.

I am trying to figure out the following problem. I am trying to evaluate the expectation: $E[\exp(-sX)\exp(-sY)]$, where $X$ and $Y$ are identically distributed, but correlated random variables, hence,...
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