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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### How to compute conditional expectation of a discrete random variable with respect to a continuous one?

while learning about conditional expectation $\operatorname{E}(X\mid Y)$, with $X$ and $Y$ being random variables, I read the corresponding Wikipedia page, where explicit formulas are given for computation of the cases where $X$ and $Y$ are both…

n_flanders

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### On the uniqueness of conditional expectation

In his probability book Bauer proves the following
Theorem. Let $X$ be an extended real-valued integrable random variable on a probability space $(\Omega,\mathcal{A},P)$, and let $\mathcal{C}$ be a sub-$\sigma$-algebra of $\mathcal{A}$. Then there…

Alphie

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### Want to compute $E(\sum_{i = 1}^n X_i^2 | \sum_{i = 1}^n X_i = t)$ for a random sample

I have $X_1, X_2, \dots, X_n$ an iid sample. My idea was to use that $\sum_{i = 1}^n X_i^2 = (\sum_{i = 1}^n X_i)^2 - \sum_{i\neq j} X_iX_j$.
Thus $$E(\sum_{i = 1}^n X_i^2 | \sum_{i = 1}^n X_i = t) = t^2 - E( \sum_{i\neq j} X_iX_j| \sum_{i = 1}^n…

Anwesha Chakravarti

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### Conditional Expectation of Square of a Gaussian

Consider two Gaussian random variables $X\sim N(0,\sigma_X^2)$ and $Y\sim N(0,\sigma_Y^2)$ with known $E[XY] = \sigma_{XY}$. The question is how to find $E[X^2\mid Y]$ assuming $X$ and $Y$ are jointly Gaussian.
My approach: I thought of the…

Mehran

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### The expected value of the lower of two values in a uniforme distribuition in [0,1], one of which is known to exceed $\frac{1}{2}$

I am reading a paper about Auction Theory, named "Auctions Versus Negotiations" by Jeremy Bulow and Paul Klemperer - 1996. Anyway, I just saw two affirmations about probabilities that I don't know exactly how to prove.
$\underline{\textbf{First}}$: …

Luiz Peres

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### If $(Y_n)$ is iid, then $Z_n:=\sum_{i=1}^nY_i$ is Markov

Let $(\Omega,\mathcal A,\operatorname P)$ be a probability space, $(\mathcal F_n)_{n\in\mathbb N_0}$ be a filtration on $(\Omega,\mathcal A,\operatorname P)$, $E$ be a $\mathbb R$-Banach space, $(Y_n)_{n\in\mathbb N}$ be an $E$-valued $(\mathcal…

0xbadf00d

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### Conditional expection of some lognormal process

Suppose I have the following process
$$y_{t+1} = y_t \exp (\varepsilon_{t+1})$$
where $\varepsilon_{t+1}\sim _{iid} N(\frac{\sigma v^2 }{2},v^2 )$, $v> 0$. I would like to calculate $E[y_{t+1}|y_t]$
For this, I apply conditional expectation in the…

Fam

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### Measurability of Conditional Expectation wrt Stopping Time $\sigma$-Algebra

Suppose that $\sigma$ and $\tau$ are stopping times. Is it true that $\mathbb E[X_\tau\mid\mathcal F_{\sigma}]$ is $\mathcal F_{\sigma\wedge\tau}$-measurable? Here $\mathcal F_\sigma$ and $\mathcal F_{\sigma\wedge\tau}$ are the stopping time…

Syd Amerikaner

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### Sum of sub-gaussian random variables

1 - If X is $\sigma_1$-subgaussian and Y is $\sigma_2$-subgaussian (not necessarily independent), then is it true that X+Y is $(\sigma_1 + \sigma_2)$-subgaussian?
I got $\sqrt{2 \sigma_1^2 + 2 \sigma_2^2}$-subgaussian by applying Cauchy-Schwarz…

user2998690

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### Expected value of $E[Z^2 | Z^3]$

Assume we have a probability space with $\Omega = [-\frac{1}{2}, \frac{1}{2}], \; \mathcal{F} = \mathcal{B}([-\frac{1}{2}, \frac{1}{2}])$, P being the Lesbegue measure and two random variables
given by
$$
X(\omega) = \omega^2 \; and \; Y(\omega) =…

Thomas

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### Doob-Dynkin Lemma, Kallenberg proof issue

In Foundations of Modern Probability, Kallenberg, (2nd edition) page 7, lemma 1.13 the following proof is given:
I can understand how the proof goes through if we assume that the measure space $(S,\pmb{S})$ is a standard Borel space (that is, a…

porridgemathematics

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### Evaluating conditional expectation w.r.t. an inequality $\mathbb{E}[X\vert X>Y]$

Let $X\in \mathcal{N}(0,1)$, $Y\in \mathcal{N}(0,1)$, and are independent. Show that
\begin{align}
\mathbb{E}[X\vert X>Y] = \frac{1}{\sqrt{\pi}}.
\end{align}
I am not sure how to handle the inequality inside the conditional. So I did the following.…

SimpleProgrammer

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### Expected 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 one. I'd like to solve it using conditional…

Alex

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### How can we explain the fact that $\mathbb E\left( {\mathbb E\left( {X|Y} \right)} \right) = \mathbb E(X)$?

We know that the following property conditional expectation holds, assuming $\mathbb E[|X|]<\infty$:
$\mathbb E\left( {\mathbb E\left( {X|Y} \right)} \right) = \mathbb E(X)$
Could anyone give me some intuition into this? why when we take the…

user830472

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### Bounding the expectation of a truncated Poission random variable

Let $\mu>0$, $k\in\mathbb N^+$, and consider a random variable $Y\sim\mbox{Poisson}(\mu)$.
I am interested in (upper) bounding $\mathbb E[Y \mid Y \ge k]$, ideally with a closed-form expression.
Notice that we have that $\mathbb E[Y \mid Y \ge…

M A

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