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]
4438 questions
154
votes
8 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
- 7,185
47
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…
user1120
40
votes
4 answers
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
- 2,015
30
votes
8 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
- 5,580
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
- 6,381
22
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
- 6,571
20
votes
4 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
- 393
- 1
- 2
- 7
19
votes
3 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
- 2,142
- 21
- 41
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
- 2,268
16
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
- 329
15
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
- 243
- 1
- 7
15
votes
4 answers
Conditional expectation given an event is equivalent to conditional expectation given the sigma algebra generated by the event
This problem is motivated by my self study of Cinlar's "Probability and Stochastics", it is Exercise 1.26 in chapter 4 (on conditioning).
The exercise goes as follows: Let H be an event and let $\mathcal{F} = \sigma H = \{\emptyset, H, H^c,…
Olorun
- 1,621
14
votes
2 answers
What is the intuition behind conditional expectation in a measure-theoretic treatment of probability?
What is the intuition behind conditional expectation in a measure-theoretic sense, as opposed to a non-measure-theoretic treatment?
You may assume I know:
what a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ refers to
probability without…
Clarinetist
- 20,278
- 10
- 72
- 137
14
votes
1 answer
Conditional expectation of product of conditionally independent random variables
I would like to show the following statement using the general definition of conditional expectation. I believe it is true as it was also pointed out in other posts.
Let $X,Y$ be conditionally independent random variables w.r.t a sigma algebra…
user401479
- 175
14
votes
3 answers
Conditional expectation of a joint normal distribution
Let $X_1, X_2$ be jointly normal $N(\mu, \Sigma)$.
I know that in general, $\mathbb{E}[X_2|X_1]$ can be computed by integrating the conditional density, but in the case of jointly normal variables, it suffices to do a linear…
user357269
- 765