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Seeking Clarification on Notation about Conditional Expectation
I am currently studying "Plane Answers to Complex Questions: The Theory of Linear Models" by Ronald Christensen, and I have encountered Proposition 6.3.2 on page 133. The proposition is stated as...
View ArticleConditional Expectation of a product of r.v.
I am exploring a work in probability theory that states $$E[U1U2 | V1, V2] = E[U1| V1]E[ U2 | V2]$$ for independent pairs of random variables $(U1,V1)$ and $(U2,V2)$. I understand the intuition behind...
View ArticleGeneral Expression for the t -th difference of conditional means
In econometrics, it is common to work with the difference-in-differences of conditional means. For example, let $Y$ denote a variable of interest and $X_{1}$ and $X_{2}$ denote binary regressors. The...
View ArticleFinding fW|U(w, u) when W depends on U? [closed]
I've been trying to answer a stat question for over a day now and haven't been able to find anything on the internet. The question is as follows:Suppose that $U$ and $V$ are independent random...
View ArticleExistence of a conditional expectation construction for metric spaces
Following the construction of the conditional expectation for real valued random variables, I wondered if it can be 'generalized' to metric spaces as follows. Let $(\mathbb{X}, d)$ be a metric space,...
View ArticleExpressing the probability of a random vector being in a random set as a...
Let $X$ be a random vector in $\mathbb{R}^n$ and let $\mathbb{P}_X$ be its distribution. Let $\{A_y : y \in \mathbb{R}^m\}$ be some collection of Borel subsets of $\mathbb{R}^n$ and let $Y$ be a random...
View ArticleDo I need additional assumptions for this equality to hold?
Suppose I have three random variables $X_1,X_2, V$, and I want the following condition to hold:$E[X_1^2|X_2<V<X_1,X_1,X_2]=E[X_1^2|X_1]=h(X_1)$,i.e., I want conditioning varibles $V$ and $X_2$ to...
View ArticleConditional expectation with respect to union of two sigma-algebra
Let $\mathscr{F}_1$ and $\mathscr{F}_2$ be two independent (sub)sigma-algebra,$X$ be a random variable with $\mathbb{E}X<\infty$.What we want to find is the relationship between...
View ArticleTerminating Sequence expected length
I was preparing for a quant interview and I came across a puzzle on QuantGuide(named Sequence Terminator):A fair 6−sided die is rolled repetitively, forming a sequence of values, under the following...
View ArticleConditional expectation of $Y \le X$ equals CDF of $Y$?
Potential proposition:I am new to studying the measure-theoretic foundations of probability. I think I have come across this result when computing probabilities/expectations.Let $X, Y$ be independent...
View ArticleRate of convergence of conditional expectation
Let $X_{0}\sim N(0,1)$, $X_{1}$ another RV independent of $X_{0}$.Let $X_{t}=(1-t)X_{0}+tX_{1}.$ We know that as $t\to1,$$\mathbb{E}[X_{0}|X_{t}]\to\mathbb{E}[X_{0}|X_{1}]=E[X_{0}]=0.$Can we determine...
View ArticleExpected number of coin flip to get HTT by conditioning
What is the expected number of (fair) coin flips to get a sequence HTT? I know similar questions have been asked before and that the answer should be $8$, but I can't seem to get my head around this...
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Am I computing conditional expectations correctly?
I am building a working model and would like to know whether my computations so far look correct.I am working with a generally bivariate distribution of two variables $\displaystyle X\sim [ 0,1]$,...
View ArticleEvaluating $\mathbb{E}[\exp(\xi_{k-1}(B_t-B_s))|\mathcal{F}_s]$ for an...
I'm reading through some notes for stochastic calculus and the author says the following:Since $\xi_{k-1}\in \mathcal{F}_{t_{k-1}}\subseteq \mathcal{F}_s$ and$B_t-B_s$ is independent of...
View ArticleCondition on more than one random variables or sigma-algebra
The first question is:Assume we have four random variables $A,B,C,D$, each one is a positive random variable with support $\text{supp}X$ but they are not independent. Given that if $A,B$ are given, the...
View ArticleFind the conditional expectation $E[X \mid X \leq p]$
Let $X$ be a discrete random variable that can take values from 1 to $n$, where $n$ is a large fixed number and also let $p\ll n$ be a fixed number. I am trying to find the expectation $$E[X \mid X...
View ArticleConditional bakery problem
This question is from QuantGuide(Bakery Boxes):A bakery manager uniformly at random selects an integer k between 1 and 4, inclusive. He then chooses from k distinct desert types at his shop to create a...
View ArticleProving the Conditional Dominated Convergence Theorem Using Classical...
I'm currently working on proving the Conditional Dominated Convergence Theorem and could use some guidance. The theorem states that for a sequence of positive random variables $(X_n)_{n\in\mathbb{N}}$,...
View ArticleOptimal Strategy for a Combinatorial Game with Asymmetric Information:...
Bob and Alice are managing a team of warriors, Alice ’s team consist of warriors with strength 3, 4, 6, 7 and Bob has 4 warriors of strength 9,8,4,2 respectively.If a warrior with strength x fights...
View ArticleUnder what assumptions $Y_n \xrightarrow{L^1} 0$ as $n \to \infty\ $?
Let $\{X_n, U_n\ |\ n \geq 1 \}$ be a collection of independent random variables such that $X_n$'s are iid random variables following standard Cauchy distribution and $U_n$'s are independent random...
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