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.

3615 questions
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Intuition behind Conditional Expectation

I'm struggling with the concept of conditional expectation. First of all, if you have a link to any explanation that goes beyond showing that it is a generalization of elementary intuitive concepts, please let me know. Let me get more specific. Let…
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Intuitive explanation of the tower property of conditional expectation

I understand how to define conditional expectation and how to prove that it exists. Further, I think I understand what conditional expectation means intuitively. I can also prove the tower property, that is if $X$ and $Y$ are random variables (or…
JT_NL
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If $E[X|Y]=Y$ almost surely and $E[Y|X]=X$ almost surely then $X=Y$ almost surely

Assume that $X$ and $Y$ are two random variables such that $Y=E[X|Y]$ almost surely and $X= E[Y|X]$ almost surely. Prove that $X=Y$ almost surely. The hint I was given is to evaluate: $$E[X-Y;X>a,Y\leq a] + E[X-Y;X\leq a,Y\leq a]$$ which I can…
Peter
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Upper and Lower Bounds on $Var(Var(X\mid Y))$

Are there any particular properties that \begin{align*} Var(Var(X\mid Y)) \end{align*} satisfies so that we can derive any upper and lower bounds on it. For example, if we replace $Var$ with expectation we have \begin{align*} E[E[X\mid…
Boby
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Understanding The Math Behind Elchanan Mossel’s Dice Paradox

So earlier today I came across Elchanan Mossel's Dice Paradox, and I am having some trouble understanding the solution. The question is as follows: You throw a fair six-sided die until you get 6. What is the expected number of throws (including…
WaveX
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Fubini's theorem for conditional expectations

I need to prove that if $E \int_a^b |X_u|\,du = \int_a^b E|X_u|\,du$ is finite then: $$E\left[\left.\int_a^b X_u\,du \;\right|\; \mathcal{G}\right] = \int_a^b E[X_u \mid \mathcal{G}]\,du.$$ I just dont have any idea how to approach this problem.
17
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Four coins with reflip problem?

I came across the following problem today. Flip four coins. For every head, you get $\$1$. You may reflip one coin after the four flips. Calculate the expected returns. I know that the expected value without the extra flip is $\$2$. However, I am…
16
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2 answers

Conditional expectation equals random variable almost sure

Let $X$ be in $\mathfrak{L}^1(\Omega,\mathfrak{F},P)$ and $\mathfrak{G}\subset \mathfrak{F}$. Prove that if $X$ and $E(X|\mathfrak{G})$ have same distribution, then they are equal almost surely. I know what I have to show, that $X$ is $\mathfrak{G}$…
14
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Fundamental Theorem of Poker

I've been doing an investigation into the mathematics behind poker, and I have stumbled upon this theorem called 'The Fundamental Theorem of Poker', which is as follows: "Every time you play a hand differently from the way you would have played…
13
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Independence and conditional expectation

So, it's pretty clear that for independent $X,Y\in L_1(P)$ (with $E(X|Y)=E(X|\sigma(Y))$), we have $E(X|Y)=E(X)$. It is also quite easy to construct an example (for instance, $X=Y=1$) which shows that $E(X|Y)=E(X)$, does not imply independence of…
13
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3 answers

Expected number of die rolls to get 6 given that all rolls are even.

A fair 6-sided die is rolled repeatedly in till a 6 is obtained. Find the expected number of rolls conditioned on the event that none of the rolls yielded an odd number I had tried to figure out what will be the conditional distribution of…
user561527
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How much are you willing to pay for this treasure chest game?

I was given an interesting problem that comes in two parts. In front of you is a treasure chest containing \$1000 with a 6-digit combination lock. You have to pay a constant amount for each time you change a digit. What is the maximum amount…
user107224
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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 $$\bar X=\frac{1}{n}\sum_{i=1}^n X_i\qquad,\qquad S^2=\frac{1}{n-1}\sum_{i=1}^n(X_i-\bar X)^2$$ as the sample mean and sample variance…
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What is $E(X\mid X>c)$ in terms of $P(X>c)$?

What is $E(X\mid X>c)$ in terms of $P(X>c)$? I've seen conditional probability/expectation before with respect to another random variable but not to the variable itself. How would I go about interpreting this?
DumbQuestion
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Asymptotic behavior of piecewise recursive random variable.

I have sequence of random variables defined by the following recursion: $$X_{n+1} = X_n+\begin{cases} \alpha(S_n - X_n), \text{ if } S_n > X_n \\ \beta(S_n - X_n), \text{ if } S_n < X_n, \end{cases}$$ where $0<\beta < \alpha <1$ are constants,…
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