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

**123**

votes

**6**answers

### 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…

Stefan

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**39**

votes

**8**answers

### 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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**32**

votes

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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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**24**

votes

**1**answer

### 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

- 5,407
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**19**

votes

**4**answers

### 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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**18**

votes

**1**answer

### 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.

luka5z

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**17**

votes

**6**answers

### 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…

user107224

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**16**

votes

**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 $\mathfrak{G}$…

Marc

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**14**

votes

**2**answers

### 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…

Mildwood

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- 1
- 7

**13**

votes

**1**answer

### 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…

user73048

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- 10

**13**

votes

**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

- 311
- 2
- 7

**12**

votes

**3**answers

### 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

- 2,128
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- 22

**12**

votes

**1**answer

### 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…

StubbornAtom

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**11**

votes

**2**answers

### 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

- 193
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**11**

votes

**2**answers

### 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,…

dezdichado

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