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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13 views

How to prove that $\forall a\in\mathbb R^+: \frac{\int_a^\infty x\cdot e^{-x^2/2}dx}{\int_a^\infty e^{-x^2/2}dx}\le a + \sqrt{2/\pi}$?

Let $a\in\mathbb R^+$ and $X\sim\mathcal N(0,1)$. I want to prove that $\mathbb E[X | X\ge a] = \frac{\int_a^\infty x\cdot e^{-x^2/2}dx}{\int_a^\infty e^{-x^2/2}dx}\le a + \sqrt{2/\pi}$. Any ideas/...
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16 views

Limit of conditional expectation of uniform random variable

If $\theta$ ~ $U(0,1)$ (Uniform random variable taking values between $0$ and $1$), and conditioned on $\theta$, $X_1,X_2,..$ are identically distributed independent random variables each having ...
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45 views

Intuition behind conditioning to events with probability zero

What’s the intuition behind why the conditional expectation w.r.t. a $\sigma$-algebra allows us to condition to events with zero probability? For example, let’s say we have two continuous random ...
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21 views

Conditional Expected value of Binomial Distribution

The expected value of a binomial distribution, $B(k, n, p)$ is $np$. How do I go about calculating the conditional expectation, given that $k>0$, $E(B(k, n, p) | k>0)$? I assume the answer is ...
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15 views

Understanding proof of discrete optimal sampling theorem

Let $X = \{X_n\}_{n=0}^{\infty}$ be a closable submartingale. Then, for any stopping time $τ, X_τ$ is integrable and, for another stopping time $σ$, $E[X_τ |\mathcal{F}_σ ]\ge X_{σ∧τ}$ , P-a.s. The ...
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63 views

Direct way to check if a random variable is $W_T$ measurable

Inspired by this: Calculate stochastic integral $\int_0^T s^2 W_s dW_s$ , I was asking myself this question: Given a stochastic integral: $I=\int_0^T f(W_s) ds$ is there a direct way to check if it is ...
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Can we simplify this expectation of ratio of expectations?

Let's say we have two random variables $O$ and $Z$ and two functions $f$ and $g$, and we are interested in the expectation \begin{align} \mathbb{E}_{p(O)}\frac{\mathbb{E}_{p(Z|O)}f(Z)}{\mathbb{E}_{p(Z|...
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15 views

Characterization of submartingale.

I am trying to prove the following are equivalent $(X_t,\mathcal{F}_t)$ is a submartingale, i.e. $E[X_t|\mathcal{F}_s]\ge X_s\;\forall t\ge s$ $\int_A X_t dP\ge \int_A X_s dP, \;\;\;\forall A\in\...
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Inference On a Selectively Revealed Sample

I think this question may be related to cryptography, so I may have the wrong stack exchange, but I am not really sure. Suppose there are two people Sam and Pam. Suppose we have a distribution, a set ...
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33 views

Joint cdf and conditional expectation problem

Suppose $X, Y$ are continuous RVs with joint pdf $f(x, y) = 0.5$ for $0\leq x\leq y\leq 2$ and $f(x, y) = 0$ otherwise. (i) Find the cdf of $Y$. (ii) Compute $P(X < 0.5 | Y = 1.5)$. Are $X$ and $Y$ ...
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44 views

Show that $X_t:=\mathbb{E}[Y|\mathcal{F_t}] $ is a martingale

I have this exercise about martingales: "Let $Y$ be a random variable with $\mathbb{E}(|Y|)<\infty$ and let $\mathbb{F}$ be a filtration as well as $X_t:=\mathbb{E}[Y|\mathcal{F_t}] $ for all $...
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45 views

Expected value of sin of sum of n random angles

Consider the following problem. Let $θ_1,θ_2,...θ_n ∈[0, \frac{π}{2}]$ be independent and uniformly distributed variables. Find $E[sin(θ_1 + ... + θ_n)].$ I was able to solve for $n=1$ (of course), ...
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27 views

Conditional density probability function [closed]

We can choose one of two investment opportunities: (A) has return £X with density fx, (B) has return £Y with density fy. We choose (A) with probability p and (B) with probability (1-p). The return on ...
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1answer
43 views

Expected Number of Moves to Fall Off Table

There is a 20cm long table. You start 3 cm from the left. Each move to the left or to the right is 10 cm with equal probability. If you reach the edge of the table you fall. In how many moves do you ...
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52 views

Calculate the percentage reduction on the variance of the claim payment

The amount of a claim that a car insurance company pays out follows an exponential distribution. By imposing a deductible of d, the insurance company reduces the expected claim payment by 10%. ...
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19 views

Applying the forward equation on a conditional expectation of a continuous Markov Chain

Suppose $X$ is a homogeneous,continuous Markov chain taking values in $\mathbb{N_0}$ with $Q$-matrix given by : $q_{n,n+1}=\lambda n + \mu$ and $q_{n,n}=-q_{n,n+1}$. Given the mapping, for $k\in \...
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42 views

Integral form of a conditional expectation

I have seen this line in a computation and I wonder if that makes sense : $\mathbb{E} (X | Y) = \int \mathbb{E} (X | Y = y) dP(Y = y)$ where P is a probability measure, $X,Y$ are random variables, so ...
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22 views

How does conditional expectation tell the average of r.v. X on the union of some basic events?

Let $X\in L^1$ be a $\mathcal F$-measurable random variable and $\mathcal G$ be a sub σ-algebra of $\mathcal F$. We say a $\mathcal G$-measurable random variable $\mathbb E[X|\mathcal G]\in L^1$ is ...
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conditional expected number of success interpretation

Suppose $f(k,n,\rho)$ is the pmf for the binomial distribution, then $$\sum_{r=0}^nf(r,n,\rho)r$$ is the expected number of successes. My question is, how can I interpret $$\sum_{r=M}^nf(r,n,\rho)r?$$ ...
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16 views

Symmetric extension to Poisson Mass Function - Conditional Variance

Let $I=\pm1$ with equal chance. Let $Y$ be independent of I and have distribution $\textrm{Po}(\lambda)$. Consider $\displaystyle X\stackrel{d}{=}IY$. Use the conditional variance formula to compute $\...
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27 views

Conditional expectation equation containing martingales

Say $M_t$ is a process such that $e^{i\lambda M_t+\frac12\lambda^2t}$ is a martingale with respect to the filtration $(\mathcal{F}_t)_{t\ge 0}$. By definition $E[e^{i\lambda M_t+\frac12\lambda^2t}|\...
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Rigorous definitions of probabilistic statements in Machine Learning

In a supervised machine learning setup, one usually considers an underlying measurable space $(\Omega, \mathcal{F}, \Bbb P)$ and random vectors/variables $X:\Omega \rightarrow \Bbb R^n, Y: \Omega \...
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51 views

Problems when solving $E(X\mid Y)$

If we know $X\sim \operatorname{Pois}(\lambda)$, $Y\sim\operatorname{Pois}(\lambda_p)$: When solving $E(X\mid Y)$, based on the law of iterated expectations, $E(X) = E(E(X\mid Y)) =\lambda$. And we ...
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18 views

how is the formula for linear regression connected to the fact that the conditional mean is the optimal estimator?

I understand that given some explanatory variables $X_i$, the best prediction in the sense of minimizing the least-squares expected error for the dependent variable Y is the conditional mean. But I ...
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24 views

Conditional expectation of this stochastic process?

I'm just beginning to learn about stochastic processes and encountered this very elementary problem that confused me a bit: We toss a coin that lands on Head with probability $p$ and Tail with $q=1-p$....
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45 views

Some equation involving $cov(X,Y)$ and $E(X|Y)$

I am trying to find some math equation envolving $cov(X,Y)$ and the conditional expectation $E(X|Y)$. I'm willing to put some hypotheses. For example, if $E(X)=0$ or $E(Y)=0$, then $cov(X,Y) = E(XY)$. ...
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124 views

Decomposing $y_n$ into $E(y_n|x_n)+\epsilon_n$. Under what condition is $\epsilon_n=o_p(1)$?

Let $y_n,x_n$ denote two sequences of random variables. Define $$ y_n=\operatorname{E}(y_n\ |\ x_n)+\epsilon_n $$ Does $\operatorname{var}(y_n\ |\ x_n)=o_p(1)\implies \epsilon_n=o_p(1)$? I tried to ...
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56 views

How to prove this process to be a martingale?

Suppose we have $(X_n)$ a sequence of real positive random variables in $\mathcal L^2$ that is $\mathcal F_n$ mesurable Let: $S_n=X_1+\dots+X_n$ with $S_0=0$ $A_n=\sigma_1^2+\dots+\sigma_n^2$ with $...
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42 views

Conditional expectation $E[X|X>0]$ of standard normal $X \sim \mathcal{N}(0,1)$

Given $X \sim \mathcal{N}(0,1)$, I was wondering what the conditional expectation $E[X|X>0]$ should be? $E[X|X>0] = \int_0^{\infty} x f(x)dx = \int_0^{\infty}x \frac{1}{\sqrt{2\pi}}e^{-x^2/2} ...
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53 views

Given W is independent of X given Y and Z, $E(Y|X,Z)=E(Y|X,Z,W)$

I am not sure whether I have interpreted the statement correctly here as it is not written in form of the formula. This is from Rubin's Statistical Analysis 3rd Edition with Missing Data page 75. &...
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41 views

An incorrect application of the Rao-Blackwell theorem

The following is given as a remark in chapter 7 of Introduction to mathematical statistics Hogg and Craig, 8th edition. (It is mentioned as "Remark 7.3.1") Note:- here $Y_1$ is a sufficient ...
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133 views

Conditional expectation of $X+Y$ given $Y-X$

Consider the following joint density function $$f_{X,Y}(x,y)=e^{-y}$$ if $0<x<y$ and 0 in other case. If I want to find the following expectation $$E[X+Y|Y-X]$$ How do I calculate? My attempt is ...
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40 views

Expectation involving multiple RV's

$X$ follows $N(0,1).$ I am supposed to find $E[X\phi(X)]$, where $\phi(X)$ is the CDF of X I know that $X$~ $N(0,1)$ and $Y = \phi(X)$~$U[0,1]$ , but I am not able to find the distribution of $XY$ ...
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38 views

Limit of expectations on increasing sigma algebra converge to expectation on limit sigma algebra

Let $\mathcal{F}_t=\sigma\{B_s:s\le t\}$ be the Brownian filtration and $t_n\nearrow T>0$ and let $Y\in\mathcal{F}_T$. I am having trouble understanding why the martingale convergence theorem ...
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18 views

Expectation of a conditional distribution

I have an input, x, which is a product of two variables $x_1$ and $x_2$. I want to compute the following expected value w.r.t $x_1$: $E_{x_1}[x * log(f(z|x))]$. $f(z|x)$ is a complex non-linear ...
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50 views

First Hitting Times of Brownian Motion have independent increments.

As I read the proof of Theorem 7.4.1 in Rick Durrett: Probability: Theory and Examples: I am confused about why a step is possible. The theorem with the proof is given in the linked picture. $T_a = \...
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35 views

Conditional Expectation of a variable given a sum of two random variables

Suppose that: $x_i = z_i + v_i,~~~~~ u_i = pv_i + e_i$ where $z_i, v_i, e_i$ are independent random variables ~$N(0,1)$. I am trying to find: $E[u_i | x_i] = E[pv_i + e_i | x_i]$ $~~~~~~~~~~~~~~= E[...
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Estimated value and GMM

I have a question. We have the following model: $y_t=\delta+\phi y_{t-1}+\epsilon_t$ $\epsilon^2_t=\varpi+\alpha \epsilon^2_{t-1}+v_t$ with $|\phi|<1, \varpi>0, \alpha \geq0$ and that: E($\...
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47 views

Expected number of coins need to reach an integral value

I'm trying to validate my work for the following problem but the computer simulation is not supporting it: There are two types of coins: one worth $0.2$ and one worth $0.4$. Now suppose a person ...
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1answer
43 views

Law of total variance with i.i.d. positive continuous r.v.s.

So law of total variance states that: $$Var(Y) = E[ Var(Y | X)] + Var(E[Y|X])$$ Now it's given that $X,Y$ are positive i.i.d. continuous r.v.s. As $X,Y$ are independent, it means $E[Y|X] = E[Y]$, ...
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60 views

How to calculate $ \mathbb{E}\left[X|W=0\right] $

Let $ X\sim\mathcal{N}\left(0,1\right) $ and define $$ W=\begin{cases} 0 & X<0\\ X & X\geq0 \end{cases} = X\boldsymbol{1}_{X\geq0} $$ How can I calculate $ \mathbb{E}\left[X|W\right]$ ? ...
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Conditional expectations with special structure. Counterexample?

We have: three random variables $X_1,X_2,X_3$, three $\sigma$-fields $\mathcal{G}_1, \mathcal{G}_2, \mathcal{G}_3$, three random variables $Y_{1,2}$, $Y_{2,3}$, $Y_{3,1}$, such that: $X_1=\mathbb{E}(...
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31 views

Expectation of a normal random variable when conditioning on a correlated normal random variable being above a threshold

Suppose $X$ and $Y$ are correlated with correlation coefficient $\rho$. They are jointly normal with means $\mu_X$ and $\mu_Y$ respectively. Then what is $E[X | Y \geq T]$? Feel free to add additional ...
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85 views

Is the following conditional expectation correct?

The following is very important in my research: suppose that $U_t, V_t$ are independent real stochastic processes defined with respect to the same filtration and probability space Suppose that for $r,...
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27 views

How does one compute the total expectation of compund binomial experiments

Assume there are two binomial experiments. For example, we coin a toss n times and get k heads. If we perform second experiment k number of times (depends on first experiment), how can we compute ...
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95 views

Characterization of a joining over a common subsystem.

Given a joining measure $\lambda$ on $X\times Y$, where $(X,\mu)$ and $(Y,\nu)$ are two probability measure space, let $\lambda=\int (\lambda_y \times \delta_y)\,d\nu(y)$ be the disintegration of $\...
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49 views

Show that $\sum_{k=1}^n\frac{1}{k}(M_k-M_{k-1})$ is a martingale

This is a problem from UMD probability quals here. I'm stuck in showing the conditional expectation requirement of a martingale. The problem is Let $(\Omega,F,\{F_n\}_{n\in\mathbb{N}},P)$ be a ...
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33 views

Showing an equivalence between the martingale property and a markov property.

I really am not sure how to get a rigorous answer to the following, any help would be greatly appreciated. Let $(X_n)_{n\geq0}$ be an integrable process, taking values in a countable set $E ⊆ \mathbb{...
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28 views

pythagoran theorem for conditional probability

I would like to get a hint for the integrability of a r.v. in my solution marked in red below. The problem I'm trying to solve is exactly the one posted here, in different notation, which I saw in ...
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32 views

Sequence of random variables depending on another random variable

I am working on the following problem: Suppose that $U\sim\rm Unif[0,1]$ and consider a sequence of random variables $X_i$ (which are iid when the value of $U$ is given) with $X_i \sim \rm Ber(U)$ (i....

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