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]
4197 questions
0
votes
1 answer
Conditional expectation conditioning on conditional variable
I wanted to confirm my understanding of the conditional expectation- intuitively it is the average of R.V given information of the conditioning variables.
In particular, given random variables $D$, $S$, $X$ and considering the conditional random…
JustBlaze
- 13
- 4
0
votes
1 answer
Expectation of conditional probability
In cs229, the problem set1 of 2019 summer, question 2(e):
enter image description here
why $p(y^{(i)}=1)$ equal to $E[p(y^{(i)}=1|x^{(i)})]$ ?
doraemon
- 123
0
votes
1 answer
A question on conditional expectation involving another conditional expectation
$X$, $Y$, $Z$ are 3 random variables.
Let $g(\cdot)$ denote a conditional expectation such that $g(X,Z)=E(Y|X,Z)$
Need to show $E(Y|X, g(X,Z))$. I think $E(Y|X, g(X,Z))=E(Y|X,Z)$ and my rationale is given below:
\begin{align}
E(Y|X,…
Jason
- 13
0
votes
1 answer
Brzezniak Example 2.3
The problem asks to find $\mathbb{E}(\xi|\eta)$. But, following the same procedure as in other cases, it is necessary to determine $\sigma(\eta)$. Brzezniak' solution compute for $\mathbb{E}(\xi|\{\eta=1\})$, $\mathbb{E}(\xi|\{\eta=2\})$ and…
Parzival
- 27
0
votes
1 answer
Explicit formula to $\mathbb{E}[Y|X_1,..,X_n]$?
Let $X_1,..,X_n$ random variables i.i.d. with distribution function $F$, with $\mathbb{P}[X_i=X_j]=0$ for $i\neq j$ and $Y$ a r.v. such that $\mathbb{P}[Y=X_i]=\frac{1}{n}$. Is there a explicit formula to…
Don P.
- 313
0
votes
0 answers
Is this inequality true: $E(A \mid A\leq B) \geq E(A \mid A\leq \min(B,C))$?
Let $A,B,C$ be any (continuously distributed, if you need) random variables such that $A$ is independent of $(B,C)$ (B and C may be correlated). I want to show that (or find sufficient conditions for)
$$ E(A \mid A\leq B) \geq E(A \mid A\leq…
lee
- 11
0
votes
1 answer
Fair coin toss as example of conditional expection
I have the following problem:
A fair coin is tossed. If it shows head(tail), two(six) dice are rolled. Let S be the sum of the numbers displayed by the dice. What is the expectation of S?
So the result is $E[S] = P('head')E[S|'head'] +…
0
votes
1 answer
Conditioning on more variables than present
Let $X,Y,Z$ three mutually independent random variables. Let $f$ an arbitrary function.
Do we have the following ? $$ \mathbb{E}\left( f\left( X,Y\right) \big| X,Y,Z \right) = \mathbb{E}\left( f\left( X,Y\right) \big| X,Y \right) $$
The tower…
anonymus
- 1,408
0
votes
1 answer
Prove that from $\mathbb E(X\mid Y) \ge Y, \mathbb E(Y\mid X) \ge X$ follows that $X = Y$.
Let $X$ and $Y$ discrete random variables. Prove that from $\mathbb E(X\mid Y) \ge Y, \mathbb E(Y\mid X) \ge X$ follows that $X = Y$.
I tried to prove it by contradiction. If the statement is not true, then without loss of generality $X \lt Y.$ Then…
great_again
- 35
0
votes
1 answer
Where is the mistake in the following application of the Law of total expectation?
Let $X$ be a random variable with $P(X=1) = P(X=-1) = \frac{1}{2}$.
Using the law of total expectation, we have:
$$
0 = E(X) = E(X | X = 1) P(X=1) + E(X | X = -1)P(X=-1) =
\frac{1}{2} E(X | X=1) + \frac{1}{2}E(X | X=-1)
$$
The first term equals…
druduche
- 3
0
votes
1 answer
Does $\mathbb E[f(X,Y)\mid X=x]=\mathbb E[f(x,Y)\mid X=x]$?
Does $$\mathbb E[f(X,Y)\mid X=x]=\mathbb E[f(x,Y)\mid X=x]\ \ ?$$
I know that if $X$ and $Y$ are independent, this hold, but does it still hold if they are not independent ?
I tried as follow :
suppose first $f(x,y)=h(x)g(y)$. Then $$\mathbb…
Walace
- 710
- 4
- 12
0
votes
1 answer
Conditional Mean and Variance - Random Walk
Given the random walk : $P_t=P_{t-1} + \varepsilon_t$ where $\varepsilon_t$ is i.i.d normal with a mean $\mu$ and variance $\sigma^2$
How do I calculate the conditional mean and variance of $P_t$, given that $P_{t-1}=x$
Edit : My working is…
ThomasVDV
- 21
0
votes
2 answers
How do I find this conditional expectation?
$X$ and $Y$ are independent $U(0,1)$ random variables.
I have to find $E[X|X>Y]$.
I found it out by integrating $\int_0^1 \int_y^1 x dxdy$ to get $\frac{1}{3}$ as the answer but this is wrong according to the answer key. What is the mistake that I…
user733666
- 578
0
votes
1 answer
Where does the extra $1_{X=x_n}$ come in discrete conditional expectation?
Where does the extra $1_{X=x_n}$ come in discrete conditional expectation?
$$\mathbb{E}[Y|X=x]=\frac{\mathbb{E}[Y1_{X=x}]}{\mathbb{P}(X=x)}$$
But then I see that when $\Omega$ is partitioned into union of subsets,…
mavavilj
- 7,270
0
votes
0 answers
Conditional expectation $E[\log(1+X) \mid X>t]$
I am wondering if the following expectation can be computed in the shown manner
$$\begin{aligned}
E[\log_2(1+X) \mid X>a] &= \int_{0}^{\infty} P( \log_2(1+X)>t \mid X>a) dt \\
&= \int_{0}^{\infty} P(X>2^t-1 \mid X>a) dt \\
&= \int_{0}^{\infty}…
hgm
- 23