All Questions
Tagged with probability-theory conditional-expectation
1,937
questions
8
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2
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491
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Conceptual Issues in the Measure Theoretic Proof of Conditional Expectations (via Radon-Nikodym)
I have been looking into measure theory (from a probabilist's perspective), and I have found the proof of the existence of the conditional expectation to feel a little "glossed over" in ...
- 912
8
votes
1
answer
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Radon-Nikodym-derivative as a martingale
At the beginning of all the stuff about Girsanov theorem, we introduced the Radon-Nikodym derivative as $Z_\infty := \frac{d \mathbb{Q}}{d \mathbb{P}}\vert_{\mathcal{F}_\infty}$.
Next, we considered ...
- 728
8
votes
5
answers
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Conditional expectation of independent variables
Claim. Let $Z_1, Z_2$ be two independent and identically distributed random variables. Then we have:
$$
\mathbb E[Z_1|Z_1+Z_2] =\frac{Z_1+Z_2}{2}.
$$
Proof.
To see this, I have proceeded as follows. ...
- 1,407
8
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1
answer
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Conditional expectation of $X$ given $|X|$
Let $X$ be in integrable, with density $f$ with respect to the Lebesgue measure.
Compute the conditional expectation : $ \operatorname{E} \left[ X\, \Big|\, |X| \,\right] $
My ansatz was : $ \...
- 83
8
votes
1
answer
2k
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Conditional expectation and independence on $\sigma$-algebras and events
In many statistics papers, proofs might proceed as follows: Under the event $A$, the random variables $X$ and $Y$ are independent. (Often this means that on $A^C$, they might be dependent). Then some ...
- 2,822
8
votes
2
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199
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Find the conditional expectation $E(U\mid \min(U,1−U))$ where $U∼U[0,1]$
Assume that $U$ is a random variable on $(Ω,\mathcal{F},P)$ with $U∼U[0,1]$. Let $Y= \min(U,1−U)$. I am asked to find $E(U\mid Y)$.
I have figured out a proof for this and I want some sanity check.
My ...
- 587
8
votes
1
answer
7k
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Polya's urn (martingale)
Suppose you have an urn containing one red ball and one green ball. You draw one at random; if the ball is red, put it back in the urn with an additional red ball , otherwise put it back and add a ...
- 273
8
votes
2
answers
376
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Show that $\mathbb{E}\left(\bar{X}_{n}\mid X_{(1)},X_{(n)}\right) = \frac{X_{(1)}+X_{(n)}}{2}$
Let $X_{1},\ldots,X_{n}$ be i.i.d. $U[\alpha,\beta]$ r.v.s., and let $X_{(1)}$ denote the $\min$, and $X_{(n)}$ the $\max$. Show that
$$
\mathbb{E}\left(\overline{X}_{n}\mid X_{(1)},X_{(n)}\right) = ...
- 433
8
votes
1
answer
290
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Show that $E(X\mid Y, Z) = E(X\mid Y)$ almost surely with condition Z is independent of $(X, Y)$
$(X, Y, Z)$ is a continuous random vector and $Z$ is independent of $(X,Y)$. Prove that $E(X\mid Y, Z) = E(X\mid Y)$ almost surely.
I had been thinking this question tonight but couldn't figure out ...
- 198
8
votes
3
answers
580
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How can two seemingly identical conditional expectations have different values?
Background
Suppose that we are using a simplified spherical model of the Earth's surface with latitude $u \in (-\frac {\pi} 2, \frac {\pi} 2)$ and longitude $v \in (-\pi, \pi)$. Restricting attention ...
- 2,187
8
votes
0
answers
242
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Regular conditional probability on Polish space and absolute continuity
Let $(\Omega,\mathcal F,\mathbb P)$ is a standard Borel space (i.e. $\Omega$ is Polish and $\mathcal F = \mathcal B(\Omega)$).
Then $\mathcal F$ is separable and for every sub-sigma-algebra $\mathcal ...
- 115
7
votes
2
answers
3k
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Holder conditional inequality
we consider, on a probability space $(\Omega,\mathcal{A},P)$, two random variable $X$ and $Y$ and let $\mathcal{H} \subset \mathcal{A}$ be a $\sigma$-algebra. Let $p,q>1$ such that $\frac{1}{p}+\...
user666433
7
votes
1
answer
651
views
Rigorous definition of the conditional expectations $E(X|Y=y)$ when $P(Y=y)=0$
Let $X$ be an integrable random variable on $(\Omega, \mathfrak A, P)$.
I've learned that for an event $A$ of non-zero probability,
$$
E(X|A) = \int X(\omega) \,dP(\omega|A) = \frac{1}{P(A)}\int_A X ...
- 3,246
7
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1
answer
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Joint densities and conditional densities of sums of i.i.d. normally distributed random variables
Let $X_1,X_2,…$ be independent with the common normal density $\eta$, and $S_k= X_1+⋯+X_k$. If $m <n$ find the joint density of $(S_m,S_n)$ and the conditional density for $S_m$ given that $S_n=t$.
...
- 5,768
7
votes
4
answers
905
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Why is $E[X|X+Y] = E[Y |X+Y]$ if X,Y are i.i.d random variables
In proof of the fact that $E[X|X+Y] = \frac{X+Y}{2}$ when $X,Y$ are independent, identically distributed random variables, one uses the observation that $E[X|X+Y] = E[Y|X+Y]$ but I don't see why this ...
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