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Computing mean and variance of a compound normal distribution with $\mu$ and $\sigma^2$ being random
Given the compound probability density $Y\sim\mathcal N(\mu,\sigma^2)$ with $\mu\sim F_\mu$ and $\sigma^2\sim F_{\sigma^2}$ how do we evaluate $\mathsf E(Y)$ and $\mathsf{Var}(Y)$ in terms of $\mathsf ... probability conditional-probability conditional-expectation- 4,071
Conditional expectation on squared sum of independent random variables
Given $X$ and $Y$ independent random variables of means $0$ and variance equal to $\sigma^2$, and $Z = X + Y$, find the conditional expectation $E[Z^2|X = x]$ for any value $x$ where the conditional ... probability statistics statistical-inference conditional-expectation- 82
What’s the expectation of one binomial random variable given the sum $n$ independent but non-identical random variables
Consider $n$ independent random variables $X_1, X_2, \cdots, X_n$ where $X_i \sim B(m_i, p_i)$ is a binomial random variable with probability $p_i$. Let $S = \sum_{i=1}^n X_i$ be the sum of these n ... probability random-variables expected-value conditional-expectation- 462
Conditional Probability Distributions
Rod and Fred have reached 6-6 in their tie-break. Rod wins any point with probability R. Fred wins with probability F, where F + R = 1. (a) Let X be the number of points until the first occasion when ... probability conditional-probability conditional-expectation- 1
If $\mathbb{E}[f(Y) \mid X=x]=c$ then $\mathbb{E}[(f(Y)-c) \mathbb{1}_{x}(X)]=0$?
I am unsure of the validity of this assertion. I would appreciate it if someone could corroborate it. Suppose we have an arbitrary function $f$ of a random variable $y$ and the expectation of such a ... probability probability-theory conditional-probability self-learning conditional-expectation- 611
Conditional expectation over sum of function
I am trying to understand how conditional expectation works when it is done over a sum of a function. Such as is the case in the following gain function g with the following properties: The gain ... functions conditional-probability conditional-expectation- 25
What is the conditional probability $P(Y|X=x)$, where $Y$ is binomial with Poisson distributed $n=X$?
Question The number of patients visiting the dentist on a day follows a Poisson distribution with $\lambda= 20$. Patients can either have one or two issues, the probability of a patient having one ... expected-value conditional-probability conditional-expectation poisson-distribution binomial-distribution- 1
Change of measure in expectation of a discrete random variable w.r.t a continuous random variable.
Let us consider that I have two random variables. A discrete one, called $\alpha:\Omega\to\{1,2,3,\ldots,n\}$ and another one called $X:\Omega\to\mathbb{R}$, which is a continuous random variable. I ... measure-theory conditional-probability conditional-expectation bayes-theorem- 51
Conditional expectation of second moment when sum is given
Let $X_1,\ldots,X_n$ be iid, positive and square integrable random variables. Further let $S:=\sum_{i=1}^nX_i$. I am trying to compute $\mathbb E[X_1^2|S]$. What i got so far is $$\frac{S^2}n=\frac1n\... probability conditional-expectation- 491
Does $E[X] = E[X|X < a]P\{X < a\} + E[X|X ≥ a]P\{X ≥ a\}$?
Question. Does $E[X] = E[X|X < a]P\{X < a\} + E[X|X ≥ a]P\{X ≥ a\}$? I would like some hint on how to start this, and mostly if this affirmation is true, whichever hint would help me a lot, I ... probability expected-value conditional-expectation- 91
Iterated conditional probability notation
I'm currently self-studying Andrew Gelman's book "Bayesian Data Analysis" third edition. At the page 41, they write: $E(\tilde{y}|y)=E(E(\tilde{y}|\theta,y)|y)$ I am ok with multiple ... measure-theory definition conditional-probability conditional-expectation iterated-integrals- 343
Conditional expectation conditional on filtration
Suppose $W(t)$ is a stochastic process adapted to filtration $\mathcal{F}(t)$. That is, for each $t$, $W(t)$ is $\mathcal{F}(t)$-measurable. I want to prove this claim: $E[W(t) \mid \mathcal{F}(t)] = ... measure-theory conditional-expectation filtrations- 1,235
Why is the conditional expectation $(x,\mu_t) \mapsto E[A(Y_t)|X_t=x]$ not Lipschitz continous?
Why is the conditional expectation $(x,\mu_t) \mapsto E[A(Y_t)|X_t=x]$ not Lipschitz continous? Long Version: I have a McKean-Vlasov equation of the type $dZ_t = H(t,Z_t,\mu_t)dt+F(t,Z_t,\mu_t)dW_t$ ... probability probability-theory continuity conditional-expectation- 11
Conditioning on an event through conditioning on a sigma-algebra
Let's consider a probability space $(\Omega, \mathcal{F}, P)$. Kolmogorov's definition of conditional expectation needs a sub-sigma-algebra of $\mathcal{F}$ as a condition. Can we define the ... probability-theory conditional-expectation- 43
Conditional Expectation in Tail $\sigma$-algebra
I am working with a sequence $Z_i=G(\mathcal{F}_i)$ where $\mathcal{F}_k=(\dots,\epsilon_{-1},\epsilon_0,\epsilon_1,\dots,\epsilon_{k-2},\epsilon_{k-1},\epsilon_k)$ where each $\epsilon_i$ are iid ... probability conditional-expectation- 197
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