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.

3,476 questions
Filter by
Sorted by
Tagged with
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/...
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 ...
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 ...
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 ...
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 ...
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 ...
10 views

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|...
15 views

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), ...
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 ...
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 ...
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%. ...
19 views

27 views

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 ...
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 ...
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 $q=1-p$....
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 $cov(X,Y) = E(XY)$. ...
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 ...
56 views

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. &...
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 ...
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 ...
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 $XY$ ...
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 ...
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 ...
50 views

24 views

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 ...
85 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 ...