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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22 views

Confusion regarding the definition of a continuous time martingale

I have come across the following definition of martingale in various texts: Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space, let $T$ be a fixed positive number, and let $\mathcal{F}(t)$, $...
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39 views

Conditional of Radon-Nikodym derivatives

When I follow the outline of the conditional as a Radon Nikodym derivative I get the following result for two measures $S$ and $P$ on $\mathcal{A}$ for $\mathcal{F} \subset \mathcal{A}$, which looks ...
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22 views

Bound on Conditional Mean of Gaussian Process.

Say $X$ is a real-valued centred Gaussian process on some topological space $T$ with continuous covariance function $K\colon T\times T\to \mathbb{R}$. Now let $(t_i)$ be a sequence in $T$, $t\in T$ ...
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54 views

Law of total expectation for three variables

I am working on this question Conditional probability - need help on calculating numerator and I got into use of the law of total expectation. How using this law I can find $$ E(X_1^2X_2^4X_3^6\mid ...
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1answer
24 views

Joint Distribution of two iid Standard Gaussian variables

I've recently been asked this during an interview and I'm very curious about how to solve questions like these because I tried looking it up with almost no results. Given $X$, $Y$ that are i.i.d. ...
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1answer
51 views

Show $P[X=x| X+Z=y]>0$ where $X$ and $Z$ are independent and $Z$ is standard normal (No Bayes Rule)

Let $(X,Z)$ be two independent random variables. We assume that $X$ is discrete and $Z$ is standard normal. Let \begin{align} Y=X+Z. \end{align} We are interested in the quantity \begin{align} P[X=x|...
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57 views

Conditional probability - need help on calculating numerator

Let $X_1, X_2, X_3$ be Geometric random variables representing the number of failures before the first success with parameter $p$. Find the conditional probability $$ P(X_1^2X_2^4X_3^6| X_1+X_2+X_3=A)....
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1answer
23 views

Computing $\mathsf{E}[ \log ( \mathsf{P}[U=a \mid V] ) \mid U=a]$

Let $(U,V)$ be a pair of random variables. I am interested in the following quantity: \begin{align} \mathsf{E}[ \log ( \mathsf{P}[U=a \mid V] ) \mid U=a] \end{align} where $a$ is some fixed constant. ...
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2answers
55 views

Conditional Expectation | Independent sigma-fields and uniqueness theorem

I have problems in understanding a small part in a proof, which is, however, a really important part. Given: $X,Y,Z$ are random variables such that $\sigma(X,Y)$ and $\sigma(Z)$ are independent $h: \...
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Why is this definition of conditional distribution measurable.

Let $X, Y: (\Omega, \mathcal{A}) \rightarrow (\mathbb{R}, \mathscr{B})$ continuous random variables with density $f_{XY}$. Now let $$f_{X|Y=y}(x) := \begin{cases}\frac{f_{XY}(x,y)}{f_Y(y)} & f_Y(y)...
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$X\stackrel{d}{=}E[X|\mathcal{G}]\implies X=E[X|\mathcal{G}]$ a.s. [duplicate]

Let $X\in L^2(\Omega, \mathcal{A}, P)$ and let $\mathcal{G}\subset\mathcal{A}$ be a sub $\sigma$-algebra. Assume that $X\stackrel{d}{=}E[X|\mathcal{G}].$ Prove that $X=E[X|\mathcal{G}]$ a.s. I have ...
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1answer
48 views

Conditional Expectation for independent sigma-algebras

Let $X,Y,Z$ be random variables such that $\sigma(X,Y)$ is independent of $Z$. How can I prove that for any measurable $A \subset \mathbb{R}$ we have: $$E[\mathbb{1}_{A}(X)|\sigma(Y,Z)] = E[\mathbb{1}...
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25 views

Determining convex function from all its expectations

Suppose I have a finite alphabet $\mathcal X$ and a convex function $h:\mathcal S(\mathcal X)\to \mathbb R$ from the simplex of probability distributions on $\mathcal X$ to the reals. Let $p_X$ be a ...
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12 views

Pareto analysis and conditional expectation

In a block of car insurance business you are considering there is a 50% chance that a claim will be made during the upcoming year. Once a claim is submitted the claim size follows Pareto with ...
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38 views

Computing $E_{\mu_{k}, \Lambda_{k}}\left[ (\mathbf{x}_{n} - \mu_{k})^{T} \Lambda_{k}(\mathbf{x}_{n} - \mu_{k}) \right]$

I wish to find: $$ E_{\mu_{k}, \Lambda_{k}}\left[ (\mathbf{x}_{n} - \mu_{k})^{T} \Lambda_{k}(\mathbf{x}_{n} - \mu_{k}) \right] $$ where the pdfs are: $$ f( \mu_{k} , \Lambda_{k}) = f( \mu_{k} | \...
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1answer
16 views

Measurability on subsets | tower property of conditional expectation

I have some understanding issues following the proof of the tower property of conditional expectation. The Theorem is the following: Let $F_0, F_1$ be $\sigma$-fields with $F_0 \subseteq F_1 \subseteq ...
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1answer
34 views

Do we have the equality $E(E(X\mid \mathcal{G})\cdot Y)=E(E(Y\mid \mathcal{G})\cdot X)$? [closed]

Let $\mathcal{F},\mathcal{G}$ be a two $\sigma$-algebras such that $\mathcal{G}\subset\mathcal{F}$. Let $X,Y\in L^2(\mathcal{F})$. Do we have the following equality: $$E(E(X\mid \mathcal{G})\cdot Y)=E(...
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1answer
19 views

Defining Conditional Expectation through projections for non-square integrable rvs

This is from Asymptotic Statistics (van der Vaart) chapter 11 (problem 4). Let $X$ and $Y$ be random variable defined on the same probability space. Define $E[X|Y]$ as the measurable function $g_0$ ...
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2answers
87 views

Find the PDF of a random vector

So I have a question in probability theory that's driving me insane. I know it is easy but I can't seem to wrap my head around it. Assume we have $U_1\sim U[-1,1]$ and $U_2 \sim U[0,2]$ which are two ...
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33 views

Finding the distribution of a vector of random variables

I've encountered this question: Let (X,Y) be a jointly distributed normal vector. Find the distribution of (X, E(X|Y)) assuming Cov(X,Y) isn't singular. I'm unsure what to do - I tried using the joint ...
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1answer
32 views

Does $E[x_i\mid y_i] = 0 \implies \operatorname{cov}(x_i, y_i) = 0$?

$$E[x_i\mid y_i] = 0 \implies \operatorname{cov}(x_i, y_i) = 0\,?$$ I am wondering if the above statement holds true. The LHS is saying that given any value of $y_i$, the expected value of $x_i$ is ...
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43 views

Ratio of the expectation over the joint distribution and the expectation over the marginal distribution

I have the following expression and I am having trouble understanding it $$\frac{\int y f_{Y,X}(y,x) dy}{\int y f_Y(y) dy}$$ Where $f_{Y,X}(y,x)$ is the joint density function of random variables $X$ ...
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2answers
116 views

Expectation of $\min(X,Y)$ conditional on $\max(X,Y)$

Let $X_1,X_2$ be two independent uniformly distributed random variables on $[0,1]$. What is the expectation of $\min(X_1,X_2)$ if at least one of them is known to exceed $0.5$, i.e $\max(X_1,X_2)>0....
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1answer
26 views

How can I express this random variable in the form of a conditional expectation?

Let $X,Y:\Omega \to \mathbb{R}$ be integrable random variables, and suppose that they are independent. Define $Z:\Omega \to \mathbb{R}$ by $$Z(\omega)=E[Y|X(\omega)\geq Y].$$ Can we write $Z=E[\tilde ...
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45 views

Proof that Expected Lifetime is longer than Remaining Lifetime if the Hazard Rate is increasing.

Let $X$ be a positive, continuous random variable. Denote the density of $X$ by $f(x)$ and its CDF by $F(x)$. Let $\bar{F}(x) = 1- F(x)$ be the survival function of $X$. Given that the Hazard Rate, \...
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10 views

For $G \sim G(3,p)$ with potential edge set $\{e_1,e_2,e_3\}$, and $\omega(G)$ the size of the largest clique, what is $\mathbb{E}[\omega(G) | e_1]$?

So I'm interested in the quantity $\mathbb{E}[\omega(G) | e_1]$. By definition, this can be calculated as $$\mathbb{E}[\omega(G) | e_1] = 1 \cdot P(\omega(G) = 1 | e_1) + 2 \cdot P(\omega(G) = 2 | e_1)...
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66 views

Conditional Expectation of the sum of Bernoulli random variables

I am struggling to understand the following computation: Let $p \in [0,1]$ and $X_1, X_2, ...$ be i.i.d. Bernoulli random variables with parameter $p$. Thus, $P(X_i = 1) = p = 1 - P(X_i = 0)$. The ...
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1answer
16 views

example that conditional independence does not imply independence

can anyone help with an example for a PMF or some density that: $P(A,B|C)=P(A|C)P(B|C )$ but $P(A,B)\not=P(A)P(B)$
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1answer
37 views

Strategy to beat the casino with unlimited amount of money (Martingales)

Brzezniak and Zastawniak's book on stochastic processes shows that that there is no way to beat the casino by having a finite amount of money available: Let $(X_1,X_2,\cdots)$ be independent random ...
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19 views

expectation of product of martingales

I have the following problem. There are two martingales (they are not, in general, independent), say $S_{i,T}(m)=\sum_{t=m+1}^{m+T}e_{i,t}$ and $S_{j,T}(m)=\sum_{t=m+1}^{m+T}e_{j,t}$. The innovations/...
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1answer
76 views

Expected value of product of dependent variables.

I've just made a quick search but can't seem to find a satisfying explanation for the following: Let $X_i,...,X_n$ be $\{-1,1\}$-variables that are not necessarily independent, and $E[X_i]=0$. Then: $...
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2answers
23 views

Conditional expectation for coin toss

A fair coin is tossed repeatedly and let T be the number of tosses before two consecutive tails occur for the first time. Show that E(T | the first toss resulted in tail) = 2 + ½E(T) Well T will have ...
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1answer
14 views

Conditional Expectation Formula on discrete random Variable Using Indicator Function

Conditioned on a discrete random variable, the conditional expectation is given by the formula : $$E(X|Y=y)=\sum xp(x|Y=y)$$ However I've found another formula in Wikipedia that given an event H: $$E(...
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1answer
47 views

Shifting balls in urns that are already occupied by balls

At the time $n=0$ we place $N$ balls in $k$ urns and change this in each step as follows: We choose one of the balls evenly distributed at random (meaning: each ball is chosen with a probability of $\...
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2answers
39 views

$X_i \sim^{iid}\operatorname{Ber}(p)$ and $Y_m = \sum_{i=1}^{m}X_i$. find $E[Y_m|Y_n]$

I have a math problem regarding condition expectancy. Let there be $$X_i \sim^{iid}\operatorname{Ber}(p), Y_m = \sum_{i=1}^{m}X_i$$. Now we know that $$Y_m\sim \operatorname{Bin}(m,p), m \leq n$$ Im ...
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1answer
20 views

Concentration result for norm of a particular double sum in separable Hilbert Space

I am seeking references for a particular setting of concentration inequality. I am deliberately being vague on some assumptions (especially on the input spaces), as I am interested in any setting. Let ...
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64 views

Find $a_n$ and $b_n$, such that $M_n=a_nY_n+b_n$ defines a Martingale, when $E[Y_{n+1}|F_n]=u_nY_n+v_n$.

Let $(\Omega,\Sigma,P)$ be a propability space and $(\mathcal{F}_n)_{n\in\mathbb{N}_0}=:\mathbb{F}$ a filtration. Let $Y_0,Y_1,\dots$ be a adapted process of integrable random variables. Further let $(...
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3answers
31 views

Calculate $E(X\mid Y=y)$ and $E[YE(X\mid Y)]$ for $g(x,y)=6xy$

We know that (X,Y) have density $g(x,y)=6xy$ for $x,y \in T$ where $T=((x,y); 0 \leq x \leq 1, 0 \leq y\leq \sqrt x )$. The task is to calculate $E(X|Y=y)$ and $E[YE(X|Y)]$. Now I find joint density ...
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40 views

Density of conditional expectation r.v. example problem

Suppose $f(x,y) = (x+y)\mathbb{1}_{1 \geq x \geq 0}\mathbb{1}_{1 \geq y \geq 0}$. Find $\mathbb{E}(X|Y=y)$ and density of $V = \mathbb{E}(X|Y)$. The first one is rather easy: $$ f_y(y) = \int_0^1(x+y)\...
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1answer
24 views

A simple question about conditional expectation

If I got $$E\left(\min\left(X,Y\right)\right)$$ Why is it equal to $$E\left(\min\left(X,Y\right)\right)=E\left(\min\left(X,Y\right)\mid\min\left(X,Y\right)=X\right)P\left(X\le Y\right)+E\left(\min\...
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1answer
59 views

$\mathbb{E}(X|Y)$ and $\mathbb{P}(X \in A | Y)$ for r.v.s $X = x$ and $Y=\sin(\pi x)$ on $([0,1], \mathcal{B}_{[0,1]}, dx)$

Suppose we have r.v.s $X = x$ and $Y=\sin(\pi x)$ on $([0,1], \mathcal{B}_{[0,1]}, dx)$. what is $\sigma(Y)$? calculate $\mathbb{E}(X|Y)$ calculate $\mathbb{P}(X \in A | Y)$ for some borel set A from ...
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1answer
35 views

Question on expected waiting time calculation

A sequence of random variables, (not necessarily independent) are generated as follows. At each time $i \in [n]$, we toss a coin with head probability $\delta_i$ (these are random variables themselves)...
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1answer
44 views

Find $E[X\mid Y]$ for $y<x$

I'm having a dillemma regarding the integral of a density: Suppose: $$f_{X,Y}(x,y) = \frac{1}{2}xy\times\textbf{1}_{x\in[0,2]}\times\textbf{1}_{y\in[0,x]}$$ In order to find $E[X\mid Y]$ I know I want ...
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1answer
17 views

On the effect of random variables on the conditional expectations

How does the 'integrablity' of a random variable have impact on 'conditioning' on a random variable or vice-versa? When do we treat one random variable as a constant or just as a random variable in ...
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2answers
69 views

If $\mathbb{E}[X^2] < \infty$ and $ g : \mathbb{R} \to \mathbb{R}$ minimizes $ \mathbb{E}[(X -g(Y))^2]$, then $g(Y) = \mathbb{E}[X\mid Y]$.

Consider random variables $X$ and $Y$ with $\mathbb{E}[X^2] < \infty$. Show that if $g : \mathbb{R} \to \mathbb{R}$ is the function that minimizes $ \mathbb{E}[(X -g(Y))^2]$, then $g(Y) = \mathbb{E}...
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1answer
31 views

if $X\mid Y$ follows Bernoulli with parameter $g(Y)$ then what is $E[X]$?

The context is not important for the question but nevertheless here it is: $A$ is the adjacency matrix of a random simple graph (A is symmetric with zero diagonal and with entries in $\{0,1\}$). The ...
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1answer
48 views

Find $\mathbb{P}_{\mathcal{F}}:2^{\{x_1,x_2,\dots\}}\times\Omega\rightarrow[0,1]$ s.t. $\mathbb{P}_{\mathcal{F}}(\cdot,\omega)$ is a prop.-measure

Let $(\Omega,\Sigma,\mathbb{P})$ be a probability space, $\mathcal{F}\subseteq\Sigma$ a $\sigma$-Algbra and $X:\Omega\rightarrow\mathcal{X}$ a random variable with a countable set $\mathcal{X}=\{x_1,...
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1answer
20 views

Conditional expectation of maximum given the sum

Let $X_1,X_2$ be i.i.d. random variables with uniform law on $\{ 1, \dots N \}$. I want to compute $$ E \left[ \max \{ X_1, X_2 \} \vert X_1 + X_2 \right].$$ How do I approach this? Do I need to ...
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0answers
16 views

Condtional expectation of piecewise function of a gaussian random variable

Let $(X_t)_t \subseteq L^2(\Omega, \mathcal{F}, P)$ be a sequence of Gaussian random variables, paramterized by $X_t \sim \mathcal{N}(\mu(t), \sigma(t))$ for deterministic, bounded functions $\mu, \...
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1answer
51 views

Describe all martingales $(X_n)_{n\in\mathbb{N}}$, such that $X_n\in\{-1,0,1\}$ for all $n\in\mathbb{N}$ with an arbitrary sample space $\Omega$.

Describe all martingales $(X_n)_{n\in\mathbb{N}}$, such that $X_n\in\{-1,0,1\}$ for all $n\in\mathbb{N}$ with an arbitrary sample space $\Omega$. This Question evolved out of this Question where $\...

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