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.

# Questions tagged [conditional-expectation]

3615 questions

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### Conditional expectation with Poisson processes and continuous uniform distribution.

$N$ is a Poisson process with intensity $\lambda$. We have arrival times $T_{1}, T_{2}, \ldots, T_{N},$ independent uniform distributed events in [0,1]. Let $T=\min \left\{T_{1}, T_{2}, \ldots, T_{N}\right\}$. I'm struggling with the following…

Almendrof66

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### Borel sigma algebra on the simplex of measure

I am trying to define rigorously the conditional distribution of random variables for myself.
Suppose I have a probability space $(\Omega,\Sigma,\mathbb P)$, a measurable random variable $X$ such that $\mathcal X=\text{supp }\{X\}$ and $\mathcal…

P. Quinton

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### Brownian Motion - Closed Form Solution

Let $S_t$ be a geometric Brownian motion defined as:
$dS_t = \mu S_t dt +\sigma S_t dW_t $
Where $W_t$ is a Wiener Process or Brownian Motion, $\mu$ is the drift term and $\sigma$ is the volatility and both are constants.
Does a closed-form solution…

Alice

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### How to calculate multivariate entropy, based on multivariate conditional entropy?

The chain rule for differential entropy says
$$h(x_1, x_2, \dots, x_K) = \sum_{i=1}^K h(x_i | x_1, x_2, \dots, x_{K-1}) $$
but not sure how the (multivariate) conditional entropies in the summand (to be broken into $K$ terms additively) can even be…

develarist

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### Expectation of the reciprocal of a positive discrete random variable

Let us assume that $X \geq 0$ is a nonnegative integer valued random variable with the below mass function.
$\mathbb {P} (X = k) = \frac {2^k \, {C \choose k} \, {C - k \choose \frac {M - k} {2}}} {{2C \choose M}} \quad \left\{ \begin {array} {ll}…

Peter

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### Refference for conditional expectation

I am having problem understanding the idea of conditional expectation with respect to a sigma algebra. Is there any reference which explains the concept in detail. I will prefer something which does not go to geometric intuitions such as orthogonal…

user813799

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### taking out what is known

From randomservices.org, a central property in conditional expectation is
$$
E[r(X) E[Y|X]] = E[r(X) Y]
$$
where $X$ is a random variable taking values in $S \subseteq \mathbb{R}^n$, $Y$ is a random variable taking values in $T \subseteq…

dunno

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### Finding $E(N_{2}\mid N_{1} + N_{3})$ where $N_{t}$ is a Poisson process

$N_{t}$ is a Poisson process.
So: We can try to find it by definition:
$$
\phi (z) = E(N_{2}\mid N_{1} + N_{3} = z)
$$
And use total probability rule, but it'll just leave us with some series, which might diverge:
\begin{align}
\phi(z) &= \sum_{k =…

vowchick

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### Application of Markov property to random walk

I have the following problem: a standard Brownian motion Y(t) (starting at zero) is sampled
at discrete times $t_j=jh$ (step size $h$). Given thresholds $d_j$ the process is said to die
at time $t=t_j$ if $Y(t_j)

gcc

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### Show that this conditional expectation is not zero

I am solving exercises from time series book and I am not sure if I am doing it right.
Consider ARMA(1,1) process : $y_t = y_{t-1} + 0.5u_{t-1} + u_t$. Show that errors obtained from regressing $y_t$ on $y_{t-1}$ i.e $0.5u_{t-1} + u_t$ don't…

Markoff Chainz

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### Proof using properties of Conditional Expectation with random vectors. Prove that $E(g(Y)|X)=E(g(Z)|X)$

I have this problem with my probability homework.
I have to prove:
Let $(X,Y)$ and $(X,Z)$ be random vectors that have the same joint distribution and $g$ a measurable Borel function that verifies $E|g(X)| \leq \infty.$
Then:
$E(g(Y)|X)=E(g(Z)|X)$
I…

Darlyn LC

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### Inequality about conditional expectation

I am trying to prove the following property about conditional expectation:
Let ($\Omega, \ F,\mathbb{P} ) $ be a probability space, let $X$ and $Y$ be two integrable random variables and let $G$ be a sub $\sigma$$- algebra$ of $\ F$. Show that if…

jack

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### Does $Y \in b(X) \iff X \in c(Y)$ implies $E[g(X,Y)\mid Y \in b(X)] = E[g(X,Y) \mid X \in c(Y)]$?

Suppose I have two random variables $X$ and $Y$ that are i.i.d. according to a prob. measure $\mu$ whose support is $A$. Take a function $g:A\times A \to \mathbb{R}$ such that $E[g(X,Y)\mid Y \in b(X)]$ exists. Furthermore, we have that…

Cristian

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### Is it possible that $\mathbb P [ Y = 1 | X = x] >0$ whereas $\mathbb P [ X = x] = 0$?

This question follows my previous one here, which is about the optimal classifier $g^*$ in case $X$ follows normal distribution.
Let $X,Y$ be random variables in which
$X$ follows normal distribution.
$Y$ takes values in $\{-1,1\}$.
In…

Akira

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### The forward rate and the connection with the future price of a zero coupon bond.

In interest modelling we ahve the the simply compounded forward rate at time t over the interval $[T,S]$, where $t

user394334

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