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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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1answer
38 views

Conditional Expectation of $\mathbb E[X^4 | X]$ [on hold]

We know that $\mathbb E[X|X] = X$ but what about $\mathbb E[X^4 | X]$?
0
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1answer
22 views

Expected value, given the sum of the random variable of interest, and another RV?

Suppose we have two random variables, X and Y, which have the same distribution (Gaussian) but different parameters. We define a third RV, Z=X+Y. Of course, the mean of Z is the sum of the means of X, ...
-1
votes
0answers
20 views

Conditional expectation given RV and an event [on hold]

I am having hard time finding out what is $E(E(X|Y, Z = z)|Y)$ equal to. Somewhat intuitively makes sense the following $$E(E(X|Y, Z = z)|Y) = E(X|Y, Z = z)$$ but I am having hard time in finding good ...
-1
votes
1answer
49 views

Prove that $\forall j,k \in \{1, …, N\}, \ \forall {n \in \mathbb N}, p^n_{j,k}=\langle T^n e_k,e_j \rangle$.

Let $(\Omega, \mathcal F,\mathbb P)$ be the probability space. Let $N \in \mathbb{N^*}$ and $(X_n)_{n \in \mathbb N}$ be a sequence of random variables with values in $\{1, ..., N\}$. Let $\mathcal ...
2
votes
1answer
32 views

How to find the expected number of moves?

Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line $y = 24$. A fence is located at the horizontal line $y = 0$. On each jump Freddy ...
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votes
0answers
28 views

GATE-2018 Probability [on hold]

A class of 12 children has two more boys than girls. A group of three children are randomly picked from this class to accompany the teacher on a field trip. What is the probability that the group ...
0
votes
0answers
33 views

Derivation of the Safety Stock Formula on Wikipedia

There is a safety stock formula on wikipedia but I've never found a step by step derivation. $$ SS = z_{\alpha } \sqrt{ E(L) \sigma_{D}^{2} + (E[D])^{2}\sigma_{L}^{2}} $$ https://en.wikipedia.org/...
1
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2answers
36 views

Markov Property and FDDs

Let $X,Y$ be two discrete time $\mathbb{R}^n$-valued stochastic processes with the same finite dimensional distributions. It may be that $X,Y$ are defined on two different probability spaces. Now, if $...
8
votes
1answer
130 views

Showing $E(S^2\mid \bar X)=\bar X$ for i.i.d Poisson random variables $X_i$

Let $X_1,X_2,\ldots,X_n$ be i.i.d $\text{P}(\lambda)$ random variables where $\lambda(>0)$ is unknown. Define $$\bar X=\frac{1}{n}\sum_{i=1}^n X_i\qquad,\qquad S^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\...
3
votes
3answers
61 views

Conditional Expectation of Bernoulli R.V.

Let $X_1, X_2,\ldots, X_n$ be iid bernoulli r.v. with parameter $p$. Let $S=X_1+\cdots+X_n$ and $Y=X_1X_2$. Compute $\mathbb{E}(Y\mid S)$. I know that $\mathbb{E}(X_1\mid S) = S/n$. So If I could ...
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votes
0answers
29 views

Probability Problem on Conditional Expectation [closed]

There are $n$ components. On a rainy day, component $i$ will function with probability $p_i$; on a non-rainy day, component $i$ will function with probability $q_i$, for $i = 1,\dotsc,n$. It will rain ...
1
vote
2answers
48 views

Conditional expectation of $\mathbf1[X_1=0]\mid X_1+\cdots+X_n$ where $X_i$'s are i.i.d Poisson RVs [closed]

Let $X_1,\cdots,X_n$ be i.i.d Poisson random variables with mean $\mu$. Let $S=X_1+X_2+ \dots + X_n$. Set $Y=\mathbf{1}[X_1=0]$. Show that $$\mathbb{E}(Y\mid S)=\left(1-\frac{1}{n}\right)^S$$ $$\...
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votes
1answer
25 views

Is “Covariance is 0” equivalent to “Conditional Expectation equals Unconditional Expectation”?

In formulas: \begin{align} Cov(X,Y)=0 \quad \Leftrightarrow\quad \mathbb{E}[Y|X]=\mathbb{E}[Y] \end{align} is this true?
0
votes
1answer
25 views

Iterated expectations conditional on an event

I know that $E(E(X|Y)) = E(X)$ from the law of iterated expectations but what if we let $B$ be a subset of the real line, then is it true that $E[E[X|Y]|Y \in B]=E[X|Y \in B]$? If so, how can I prove ...
1
vote
2answers
47 views

Explaining Why the Zero Conditional Mean Assumption is Important

I am currently relearning econometrics in more depth than I had before. One thing I am trying to make sense of currently is why it is necessary for the assumption of: $$E(u\mid x)=E(u) $$ to be true (...
4
votes
1answer
44 views

Posterior mean if signal is an interval rather than a realization

Suppose that a signal or observation $s_1$ is drawn from the normal distribution $\mathcal{N}(\mu,\sigma^2)$, where $\sigma^2$ is known but $\mu$ is not. We want to estimate $\mu$ based on $s_1$. ...
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votes
0answers
62 views

How to calculate this conditional expectation?

Let $\Omega$ be $[0, 1]$ and $\mathcal{F}$ be the borel sigma-algebra restricted to $[0,1]$. Let G be the sigma-algebra generated by sets $[0, 1/5), \{1/5\}, (1/5, 1/2)$. Define random variable X: $\...
0
votes
0answers
27 views

Strong Markov Property and consequence on hitting time of Sphere

We introduced the Strong Markov Property (MVP) as follows: Let $T$ be a finite Stopping time and $(B_t)_{t\geq0}$ a Brownian motion on a filtered probability space, $F_{\infty}=\sigma(\cup F_i)$ and $...
1
vote
0answers
18 views

Difference between “given” and “sampled from” in an expectation

I'm reading a paper that states that trajectories $\tau$ are sampled from a distribution $\pi$ and they use the notation for the expectation $\mathbb{E}_{\tau \sim \pi}$ but also use $\mathbb{E}_{\tau ...
3
votes
1answer
54 views

Conditional expectation from joint distribution

I am new to probability and trying to convince myself of the correctness of the equations in this paper on factor analysis. There is a step I am missing. I'll give my understanding so far and then ...
0
votes
0answers
24 views

Dirchlet vectors conditioned on inner product being equal

Assuming that $X_1, X_2 \in \mathbb{R}^n$ are two Dirichlet vectors with parameters $\alpha, \beta \in \mathbb{R}^n$ i.e, $$ X_1 \sim \operatorname{Dirichlet}(\alpha), X_2 \sim \operatorname{...
4
votes
0answers
56 views

What is the intuition behind conditional expectation in a measure-theoretic treatment of probability?

What is the intuition behind conditional expectation in a measure-theoretic sense, as opposed to a non-measure-theoretic treatment? You may assume I know: what a probability space $(\Omega, \mathcal{...
1
vote
2answers
29 views

Conditional expectation disjoint events

Let $X$ be a random variable and $A$ and $B$ be two mutually exclusive events, is it true that: $$E(X \mid A \cup B) = \frac{P(A)E(X|A) + P(B)E(X|B)}{P(A \cup B)} $$ If so, how can I prove it?
0
votes
1answer
24 views

Conditional expectation and sampling without replacement

Suppose I have $N$ numbers: $x_1,\cdots,x_N$ such that $N$ is even and $\sum_{i=1}^N x_i = 0$. Let $X_j$ be the $j$-th number I get when I sample from $\left\{x_1,\cdots,x_N\right\}$ without ...
0
votes
2answers
32 views

The problem for conditional expectation

The joint probability density function for random variables $X$, $Y$ is given by $$f(x, y)=\begin{cases} 2(x+y) & \text{if } 0<x<y<1 \\ 0 & \text{otherwise} & \end{cases}.$$ ...
1
vote
1answer
15 views

Lindeberg condition implies L1 convergence of cond. variances

The following is taken from Dvoretzky, 1972, ASYMPTOTIC NORMALITY FOR SUMS OF DEPENDENT RANDOM VARIABLES, Equation 4.6. $$\{X_{n,k}\}_{n=0,1,...;k=0,1...,k_n}$$ is a (triangular) array of r.v.'s /w ...
0
votes
1answer
44 views

How is salary a binomial distribution?

In an online course here, the author presents a problem that salary is normally distributed, provides mean and variance. As per my understanding, when no of Bernoulli trials are sufficiently large ...
0
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4answers
39 views

Conditional expectation application

Suppose I have two non-negative r.v.s $X,Y \geq 1$ and I know $\mathbb{E}(X^2 | Y) = (Y-1) ^ 2$. Does this mean $\mathbb{E}(X | Y) = Y - 1$? Cause, if i recall right the intro to probability, I have ...
1
vote
1answer
28 views

Convergence of conditional expectations equivalence

Suppose that as sequence of random variables $\{X_n\}$ and $X$ are defined on $(\Omega,\mathcal{F},P)$ and $\mathcal{G}\subset \mathcal{F}$. I know that for any $G\in \mathcal{G}$, $$ E[f(X_n)1_G]\to ...
4
votes
3answers
379 views

How do I convert a simple algorithm to a mathematical notation?

I am trying to write a simple algorithm given below in mathematical notation. I wrote the formula up to a certain point, but I have no idea of restrictions. For example, I used the NULL statement even ...
1
vote
0answers
10 views

Stochastic process modeling point hopping

Let $(X_n)_n$ be a stochastic process (with application regards $X_n$ as a distance between two points, hence those squares) with the next property $\mathbb{E}(X_n^2 | X_{n-1}) = X_{n-1}^2 - X_{n-1} -...
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votes
1answer
27 views

Law of iterated expectations applied to a ratio

Consider the random variables $Y, Z_1, Z\equiv(Z_1,\dots,Z_n)$ with $Z_1,...,Z_n$ i.i.d. Is it true that $$\mathbb E\left(\frac{Y}{Z_1}\right)=\mathbb E\left(\frac{\mathbb E(Y\mid Z)}{Z_1} \right)$$?...
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0answers
12 views

conditional duality pairing?

I tried to read the paper my supervisor gave me that is https://arxiv.org/abs/1807.02666v1 but I have sooo many questions. My gf suggests I should write the authors, but they may know my boss and this ...
0
votes
2answers
15 views

Percent loss covered on average with a deductible

Suppose you have auto insurance with a deductible of \$200, and with no restriction on maximal payment. The probability of a loss is 0.10, and suppose that the distribution of the loss is exponential ...
0
votes
0answers
61 views

Finding the argument of a complex fraction (doing something wrong but not knowing what)

I am working on a problem, and I think that I miss something very simple but I do not get what I am doing wrong. The problem: I need to find the argument of the following function (all the ...
0
votes
3answers
25 views

Expected weight split in a sum of independent variables

Let $x,y \geq 0$ be independent random variables with the property that $E[x] = E[y]$. Can I infer that $$P(x \geq E[x] \mid x + y \geq E[x+y]) = 1/2~?$$ My heuristic reasoning is that, since $x$ ...
2
votes
1answer
111 views

How to find E[|Product of Two Gaussians|]

The Problem Let $Y \sim N(0,B)$ and $Z \sim N(0,A)$. I would like to find $$\boxed{\mathbb{E}\left[|YZ|\right]}$$ This seems like a natural question that would arise in probability theory, but I ...
1
vote
0answers
30 views

Moment Generating function with Poisson and Exponential distributed variables

what happens when I have the following MGF with the $N$ and $M$ independent: $$ E(e^{-\rho R N M}),\mbox{ with }N \sim \mbox{Poisson} (\lambda), M \sim \mbox{Exponential} \left(1/t,1/t^2\right).$$ A ...
3
votes
0answers
40 views

Integral over conditional expectation

Suppose that $(t,x)\mapsto g(t, x)$ is continuous in $t\in [0,1]$ and $\mathsf{E}\sup_t |g(t,X)|<\infty$, where $X$ is some random variable living on $(\Omega,\mathcal{H},\mathsf{P})$. Is the ...
0
votes
1answer
14 views

Calculating mutual conditional expectation

I'm having trouble getting behind the process of applying the theory to calculating specific conditional expectations. I know that this might be a simple question, but I can't see the simplicity. Let $...
1
vote
1answer
36 views

A question conditional expectation

Define $M(Z_1, \dots, Z_n)$ to mean the closed subspace of $L^2$ consisting of all random variables in $L^2$ of the form $\phi(Z_1, \dots, Z_n)$ for some Borel function $\phi: R^n \rightarrow R$. Is ...
4
votes
1answer
46 views

Calculate $E(\max (X,Y) \mid X) $ if $X$ and $Y$ are independent and exponential distributed

Let $X,Y$ be independent and both $\text{Exp}(\lambda)$ distributed. How does one calculate $$E(\max (X,Y)\mid X)\,?$$ By independence $\max (X,Y)$ is $\text{Exp}(2\lambda) $ distributed but I do ...
2
votes
0answers
31 views

Clarification of one of the properties of conditional expectations

Here is one of the properties of conditional expectations stated in "The Theory of Stochastic Processes I" by Gikhman and Skorohod (on page 33): The proof relies on the fact that there is a sequence ...
1
vote
1answer
24 views

A question about conditional expectations: $E[G(\xi)|F(\xi, \eta)]=E[G(\eta)|F(\xi, \eta)]$

$\xi$, $\eta$ are random variables, and they are independent and identically distributed. $F(x, y)$ is a Borel function such that $F(x, y) = F(y,x)$. $G$ is a Borel function such that $G(\xi)$ is ...
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votes
1answer
50 views

Expected Value from Conditional Expected Value

Here is a question I've been thinking about, but for which I couldn't find a solution. Let $X\sim Exp(\alpha)$ and $Y|X=x\sim Exp(\sqrt x)$ (Y conditional X=x). What is the expected value and ...
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votes
2answers
59 views

X and Y identically distributed implies measurability of X and Y (especially Y=$\mathbb{E}(X|\mathcal{G})$

X and Y random variables on ($\Omega$, P, $\mathcal{F}$). If Y is measurable regarding a sub-$\sigma$-algebra $\mathcal{G}$ and X and Y are identically distributed, is there a way to show that X is $\...
1
vote
1answer
22 views

Conditional expectation of two dependent rvs which consist of independent uniformly distributed rvs

I am self-thaught in advanced probability and one of the exercises which I found is: "Let $\xi$ and $\eta$ be independent rvs uniformly distributed on $(0,1)$. Let $X=\xi\eta$ and $Y=\xi/\eta$. ...
0
votes
2answers
50 views

If $Y|X=x\sim N(x,x^2)$ and $X\sim U(0,1)$ is it true that $EX=E(Y|X)$? [closed]

I tried to apply the definition of conditional expectation to it but with no success. Any hint, anyone?
1
vote
0answers
17 views

Trouble following proof of Rule of Iterated Expectations

My text has the following definition of expectation: $$\mathbb{E}(X) = \int x dF(x) = \begin{cases} \sum_x xf(x) & \text{ if X is discrete } \\ \int xf(x)dx & \text{ if X is continuous } \end{...
1
vote
2answers
44 views

“Plugging into” conditional probability - why does it work?

I think the best way to ask this question is using an example. Let $X$ be a continuous random variable and $Y$ a (not necessarily continuous) random variable that is independent of $X.$ Consider the ...