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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.

2
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0answers
21 views

proof of conditional expectation given n i.i.d. random variables

This is another question from my self-study of Hayashi's Econometrics. How do we show in mathematical proof that given: $X = \begin{bmatrix}x_{1}' \\x_{2}' \\\vdots \\x_{n}'\end{bmatrix}$ where $x_{...
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1answer
27 views

Finding conditional expectation $E[X_1 | X_2 = x_2]$

I am trying to find $E[X_1 | X_2 = x_2]$ under these premises. $X_1 \sim Unif(0,1) \space , \space X_2 \sim Unif(0,x_1)$ I was able to find that the joint pdf is $$f(x_1,x_2) = \frac{1}{x_1} \...
0
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1answer
45 views

How to estimate the parameters of a censored exponential mixture process.

I have a real world scenario that involves a process behaving as follows: there are two kinds of machines and when these machines fail, their recoveries follow exponential distributions. Let's say the ...
0
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0answers
17 views

Conditional Expectation of Gaussian matrix

Let $\boldsymbol{a} \sim \mathcal{N}(0,I)$. Then what is the expectation of the function. $E(\boldsymbol{a}\boldsymbol{a}^T| (\boldsymbol{a}^T\boldsymbol{x})^2 \leq c )$. Where $\boldsymbol{x} $ ...
0
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0answers
21 views

Proof that Polya's Urn is a Martingale

In these Wisconsin Lecture Notes on probability, they sketch that, in Polya's Urn, the fraction of green balls at a step $n$ defines a martingale. I would like to justify these steps and make them ...
1
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0answers
17 views

Third Moment of Hitting Time

We recently went through expected hitting times of markov chains in my class and were asked about computing the various moments of hitting times. As such, I'm wondering if my thinking is correct as ...
0
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2answers
39 views

Conditional expectation for iid (independent and identically distributed) random variables

Let $X_1, X_2, X_3$ be iid $N(\mu,\sigma^2)$ random variables. Find $E[2X_1+3X_2|X_1+3X_2-X_3=4]$. Inuitively, I think this expectation will be equal to $E[4+X_1+X_3]$, hence $4+2\mu$. As $2X_1+3X_2 =...
0
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1answer
26 views

Probability given information

I am looking too see that $\mathbb{P}(X > \mathbb{E}[X|\mathscr{F}])|\mathscr{F})$ is not the same as $\mathbb{P}(X > \mathbb{E}[X])$$. In other words, some event happens and I want to calculate ...
2
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0answers
28 views

If Y is positive then show that $\{Y=+\infty\} \subset \{E[Y\mid \mathcal{G}]=+\infty\}$(Almost surely for all $\omega \in \Omega$)

If Y is positive then show that $\{Y=+\infty\} \subset \{E[Y\mid \mathcal{G}]=+\infty\}$ where $Y$ ,lives on $(\Omega,\mathcal{F},P)$ and $\mathcal{G}\subseteq \mathcal{F}$.() My attempt In order to ...
1
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1answer
22 views

Basic Question on Conditional Expectation being in $L^2$

Suppose that $X,Y$ is a random-variables in $L^2(\Omega,\mathcal{F};\mathbb{P})$, where $(\Omega,\mathcal{F};\mathbb{P})$ is a complete probability space. Then the conditional expectation $E[X|\sigma(...
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votes
1answer
45 views

Compute $\mathbb{E}(2X_1+3X_2|X_1+3X_2-X_3=4)$ for $(X_k)$ i.i.d standard normal

$X_i,i=1,2,3$ are $i.i.d$ standard normal random variables. Then $\mathbb{E}(2X_1+3X_2|X_1+3X_2-X_3=4)$=? (Source: UOH PhD Entrance 2017) I am not able to proceed too much with the problem ...
0
votes
2answers
88 views

Expectation of $X$ given $X > Y + a$

Let $X$, $Y$ be independent bounded integrable random variables and let $a$ be any constant such that $$P(X > Y + a) > 0$$ Is $$E[X | X > Y + a]$$ weakly increasing in $a$? This thread ...
2
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2answers
47 views

Find $\mathbb{E}(h(X) \mid U)$ where $h$ is measurable, $X, Y$ are independent and $U = \max(X,Y)$

Find $\mathbb{E}(h(X) \mid U)$ where $h$ is measurable, $X, Y$ are independent and $U = \max(X,Y)$. $X$, $Y$ follow both an exponential distribution with parameter $\lambda = 1$. I couldn't find a ...
0
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2answers
16 views

$\sigma$-algebra making Conditional Expectation equal to Expectation

Is there a choice of a $\sigma$-algebra which makes the conditional expectation of an random-variable $X \in L^1_{\mathbb{P}}(\mathcal{F},\mathbb{P})$, on a probability space $(\Omega,\mathcal{F},\...
1
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0answers
24 views

Conditional Expectations of maximum of rv

Hi I am trying to prove the following: Let $\{v^j\}_{j = 1}^n$ be iid random variables $v_j : \Omega \rightarrow [\underline{v}, \bar{v}]$ and let $H(\cdot)$ and $h(\cdot)$ be their cdf and pdf, ...
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0answers
10 views

Conditional expectation of a bounded harmonic function

Let $G$ be a discrete group. Consider $G^{\mathbb{N}}$, the space of all sequences in $G$ equipped with product $\sigma$-algebra. Let $m:G^{\mathbb{N}} \to G^{\mathbb{N}}$ be the multiplication map ...
1
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1answer
30 views

Find the conditional density function and expectation of $Y$ given $X$ when $f(x,y) = \lambda^{2}e^{-\lambda y}$ and $f(x,y) = xe^{-x(y+1)}$

Find the conditional density function and expectation of $Y$ given $X$ when they have joint density function: (a) $f(x,y) = \lambda^{2}e^{-\lambda y}\quad\text{for}\quad 0\leq x < y < +\infty$ ...
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2answers
45 views

What is the expected delay between the grey buses?

Four yellow buses and two grey buses that could be in any order (with equal probability) are traveling together with the probability of a delay of at least $\mathit{t}$ seconds between any two ...
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0answers
21 views

Prove the linearity of the conditional expectation.

The question I am working on is: Prove the linearity of the conditional expectation: $E[f_1(x)y+f_2(x)|x]=f_1(x)E[y|x]+f_2(x)$ What I tried: Looking up properties of expected values, similar ...
2
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2answers
50 views

without computing it, show that $I = \mathbb{E}[e^{XY} |X] \geq 1$

$X, Y$ are two independent $\mathcal{N}(0,1)$ random variables this question was a follow up question of this one I honestly don't know if that series of equivalences could be of any help. my ...
2
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0answers
28 views

Understanding Conditional Expectation and relation to Crossed Product

Let $\mathcal{A}$ be a unital $\Gamma$-$C^*$-algebra. Then one can form the reduced crossed product $C^*$-algebra $\mathcal{A}\rtimes_r\Gamma$. The reduced crossed product comes equipped with a ...
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0answers
26 views

Expected number of rolls of die until gcd/lcm exceeds certain value?

I was presented with the following problem: What is the expected number of rolls of a $n$-sided die until i) the lcm of the rolls exceeds $k$? ii) the gcd of the rolls exceeds $k$? Is this ...
2
votes
1answer
38 views

Conditional expectation of asymptotically independent random variables

Suppose that $W_n \to W_{\infty}$ a.s. where $W_{\infty}$ is independent of random variable $V$. Moreover, suppose that $E[|V|]<\infty$. Is it true that \begin{align} \lim_{n \to\infty} E[V|...
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0answers
22 views

conditional distribution of $X$ given $\min(X,\alpha)$

let $\alpha \in \mathbb{R}.$ So I want to find a conditional distribution for $X$ given $Y=\min(X,\alpha).$ I know that for all $B \in \mathcal{B}(\mathbb{R})$ $$P_Y(B)=\int_B 1_{]-\infty, \alpha]}(x)...
0
votes
1answer
16 views

Higher Order Conditional Expectation

This may be a very silly question, but does $E[y_t| I_{t-1}] \neq 0$, where $E[y_t|I_{t-1}]$ is the expectation of $y_t$ conditional on the information available at time $t-1$, implies $E[y^2_t| I_{t-...
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0answers
10 views

Quantile decomposition [proof verification]

Problem: We have (real-valued) stochastic variables $\{r_i\}_{i=1}^N$ and weights $\{w_i\}_{i=1}^N$ satisfying $0 \leq w_i \leq 1$ and $\sum_{i=1}^N w_i = 1$. We define a stochastic variable $$r_p = \...
3
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1answer
43 views

On calculating sigma algebras generated by specific functions.

I started (again!) with the intention to build an interesting example of a computation of conditional expectation with respect to $\sigma(X) $ when $X $ is not a step function. My first example, ...
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0answers
24 views

$\mathbb{E}[f(X,Y)|Y]=\mathbb{E}[f(X,Y)]$ for all bounded measurable $f$

Let $X,Y$ be independent rv's. For $f:\mathbb{R}^2\to\mathbb{R}$ measurable we have $f_X:\mathbb{R}\to\mathbb{R}$ where $$f_X(y)= \left\{ \begin{array}{lr} \mathbb{E}[f(X,y)] & : \...
0
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1answer
32 views

Can you explain this counterintuitive conditional expectation result intuitively?

Consider the following experiment. We throw a three-sided die with sides $1$, $2$ and $3$ infinitely many times. Let $T_i$ denote the outcome of the $i$'th throw. Define $N:=\min\{i:T_i\neq1\}$. Let $...
0
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1answer
49 views

Non-trivial examples of $E[X|X^2] = X$ and $E[Y|Y^2] = 0$.

I want to find a few examples of non-trivial random variables $X, Y$ such that $E[X|X^2] = X$ and $E[Y|Y^2] = 0$. From what i gather for $E[X|X^2] = X$ i need to find a function such that if $X=u$ ...
1
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1answer
21 views

Limit of martingale is not $\Sigma_n$-measurable

Let $U_n\sim$ Unif$(-1,1)$ and $\Sigma_n=\sigma(U_1,...,U_n)$. Define $X_0=0$ and $$X_n=X_{n-1}+(1-|X_{n-1}|)U_n.$$ Given that for the limit $X$ of $X_n$ we have $\mathbb{P}(X=-1)=\mathbb{P}(X=1)=\...
0
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1answer
15 views

Prove conditional expectation of standard normal random variables

Let $X_{i},i=1,2,...$ be a sequence of independent identically standard normally distributed random variables. Let $\{\mathcal{F}_{n},n\in\mathbb{N}\}$ be the natural filtration and $S_{n}=\sum^{n}_{i=...
0
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0answers
76 views

Given $X\sim\text{Binomial}(m,p)$ and $Y\sim\text{Binomial}(n,p)$, calculate $\textbf{P}(X = x | X + Y = s)$

Let $X$ and $Y$ be independent RV's such that $X\sim\text{Binomial}(m,p)$ and $Y\sim\text{Binomial}(n,p)$. Determine (a) $\textbf{P}(X = x \mid X+Y = s)$ (b) $\textbf{E}(X\mid X + Y)$ and $\textbf{V}...
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1answer
23 views

A problem involving geometric distributions and conditional expectation

Take into account an infinity sequence of independent tosses of an unbiased die. Let $X$ be the number of tosses necessary to obtain a five and $Y$ the number of tosses necessary in order to obtain a ...
1
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1answer
29 views

Why is the conditional expectation of an $L^{p}$-function again in $L^{p}$?

Let $(\Omega, \mathcal{A},P)$ be a probability space and let $X,Y\colon \Omega \rightarrow \mathbb{R}$ be random variables. Furthermore, let $Y$ be $p$-integrable. Then why is the conditional ...
3
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2answers
320 views

If the expectation and variance of $X$ are both not affected by $Y$, and vice versa, then must $X$ and $Y$ be independent?

I know that if $\mathbb{E}[X]=\mathbb{E}[X|Y] , \mathbb{E}[Y]=\mathbb{E}[Y|X]$, $X$ and $Y$ can be dependent, for example a ‘uniform’ distribution in a unit circle. Now we add the variance, if $$\...
2
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4answers
88 views

Conditional expectation of minimum of exponential random variables

Let $X_1$ and $X_2$ be independent exponentially distributed random variables with parameter $\theta > 0$. I want to compute $\mathbb E[X_1 \wedge X_2 | X_1]$, where $X_1 \wedge X_2 := \min(X_1, ...
1
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0answers
24 views

Where is the flaw in my reasoning when computing the expectation of a Brownian motion dependent on a rate one Poisson process?

We have that $\{W_t:t\geq 0\}$ is a Brownian motion, and $\{N_t:t\geq 0\}$ is a rate one poisson process which is independent of the Brownian motion. We must show that $$\mathbb{E}[W_{N_t}^2]=t$$ I ...
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1answer
28 views

Prove E(A|G) = $1/2(1_A(\omega) + 1_A(-\omega))$, where G = $\sigma(x)$ for $x = \omega^2$

I am extremely stuck on this problem, no idea how to even get started. Any help appreciated. Let $(\Omega, F, P) = ([0, 1], B([0, 1]), \operatorname{Leb}/2)$, where Leb refers to the Lebesgue measure....
2
votes
1answer
40 views

Conditional expectation of Brownian motion given stopping-time sigma algebra

Let $W$ be a Brownian motion with filtration $(F_t)$. Let $\tau$ be a stopping time. It is well-known by the strong Markov property that the law of $W_{\tau+t}-W_\tau$ given $F_\tau$ is normal with ...
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2answers
46 views

Given a joint density function, what is the conditional expectation $E[Y|x]$?

The random variables $X$ and $Y$ have the joint density $$f_{X,Y}(x,y)= \left\{\begin{matrix}e^{-y}, \mbox{ } 0\leq x \leq y \le \infty \\ 0, \mbox{ otherwise}\end{matrix}\right.$$ Evaluate the ...
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2answers
61 views

The random variables $X$ and $Y$ have the joint density $f_{X,Y}(x,y)=…$ [duplicate]

The random variables $X$ and $Y$ have the joint density $$f_{X,Y}(x,y)= \left\{\begin{matrix}e^{-y}, \mbox{ } 0\leq x \leq y \le \infty \\ 0, \mbox{ otherwise}\end{matrix}\right.$$ Evaluate the ...
0
votes
0answers
12 views

Symmetrization of Conditional Expectation without common density

I know that the following rule for (factorizations of) conditional expectations is true: Theorem Let $X : \Omega \to \mathcal{X}=\mathbb{R}$ be an integrable, real valued random variable and let $Y:...
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votes
1answer
40 views

Conditional variance when there is uncertainity about the distribution of condition

Assume that $X \sim N(0,\sigma_x^2)$. Y has the following form \begin{align} Y &= \begin{cases} Y_1 \sim N(0, \sigma_1^2), & \text{w.p.} \quad \mu \\ Y_2 \sim N(0, \sigma_2^2), & \text{w....
1
vote
1answer
85 views

If $(κ_t)_{t≥0}$ is the transition semigroup of a continuous Markov process, is $t↦(κ_tf)(x)$ continuous for all bounded continuous $f$ and fixed $x$?

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space $(\mathcal F_t)_{t\ge0}$ be a filtration on $(\Omega,\mathcal A)$ $E$ be a metric space $(X_t)_{t\ge0}$ be an $E$-valued right-...
0
votes
2answers
17 views

$\mathbb{E}[(Y_{n+1} - {Y_n})Z] =0.$ implies $Y_n = \mathbb{E}[Y_{n+1}|H_n]$?

Suppose $Y_n = \mathbb{E}[Y|X_1 \dots X_n]$ is a Doob's Martingale sequence . let $H_n$ be $\sigma$-algebra generate by $X_1 \dots X_n$,then if for any random variable $Z \in H_n$ we have :$\mathbb{E}...
0
votes
0answers
24 views

Truncated conditional expectation of multivariate normal distribution

Assume that $y_1 = \alpha_1 s_1 + u_1$ and $y_2 = \alpha_2 s_1 + \alpha_3 s_2 + \alpha_4 u_1 + \alpha_5 u_2$ where $s_1 \sim N(0,\Sigma_1)$, $s_2 \sim N(0,\Sigma_2)$, $u_1 \sim N(0,\sigma^2)$, $u_2 \...
2
votes
1answer
72 views

Conditional expectation with a third random variable

In this post some basic steps were given as understood, and I've been trying to fill in the gaps without much success. Specifically, the problem calls for random variables $X,$ $Y,$ and $U,$ linked by ...
6
votes
3answers
72 views

Prove that $U=Y - E[Y|X]$ and $X$ are uncorrelated

Let $U = Y - E[Y|X]$. How can I prove that $U$ and $X$ are not correlated? I've been doing a lot of things but when I calculate $\text{cov}(U,X)$ I finish with $EXY - EXEY$ and not $0$ which would be ...
2
votes
1answer
48 views

How is Grönwall's inequality applied here?

Let $h\in C^1(\mathbb R)$ such that $h'$ is Lipschitz continuous and $$L\varphi:=-h'\varphi'+\varphi''\;\;\;\text{for }\varphi\in C^2(\mathbb R).$$ Now, let $(X_t)_{t\ge0}$ be the unique strong ...