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
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Expected value of a partly limited variable
I have a function: $f = \max(0,1-\frac{A}B) $, where $B$ is known, $A$ is a normal random variable $N(\mu,\sigma)$ with expected value of $\mu$ and standard deviation of $\sigma$.
What I need is a closed formula for the expected value of $f$…
guyko
- 1
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1 answer
Conditional Expected Value $E[(X-Y)^2] < \infty $
Given
$E[|X|]<\infty$, $E[|Y|]<\infty$, $E[X|\sigma(Y)]=Y$, $E[Y|\sigma(X)]=X$,
I need to show that out of $E[X^2]<\infty$ and $E[Y^2]<\infty$ follows also $E[(X-Y)^2]<\infty$.
I started with using the linearity of the expected value to my…
N. Maks
- 237
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1 answer
Conditional Expectation for a Joint PMF
Let (X; Y ) have joint mass function $P(k, n) = \frac {C*2^-k}{n}$ , for k = 1, 2, and
n = 1, 2, , k, and suitable constant C. Compute $E(X|Y = y)$.
Its easy to calculate the $P(X,Y)$ but i am getting confused in calculating $P(Y)$ also what…
Raveesh
- 135
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Conditional probability of $E[X_{1}\mid X_1+X_2+\cdots+X_{n}]$
Find conditional probability of $E[X_{1}|X_1+X_2+...+X_{n}]$ given the
fact that $X_{k}$ for all $k$ have finite expected value and all the
variables are independent and have the same distribution.
I guess the solution should follow quickly…
maq
- 669
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1 answer
Show E[E(X|Y)]=E[X]
Show E[E(X|Y)]=E[X]
Now if X and Y are independent then it is very straightforward as E(X|Y)]=E[X].
However is there a better explanation to this ? Can we prove this mathematically ?
What if X and Y are not independent ? Will it hold ?
Aviral Gupta
- 65
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1 answer
Conditional expectation and inequality
This should be relatively easy question but my math skills are rather limited.
Assume a that i is uniformly distributed between $\ [0,3] $
Also, assume that there is a price such that:
$$p \ge E[v * i | w * i \le p]$$
v and w are two unknown…
user1738753
- 103
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1 answer
Conditional distribution of iid variables
I am given several four iid random variables (say a, b, c, d), each with a distribution of N(0.2,0.01). How does the sum of these four iid RVs (i.e. Z=a+b+c+d) distribute under the condition of d=0.3? I thought this should be the same as Z's…
Shilimu
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Reminder on Conditional Expectation
I've refreshed myself on the properties of conditional expectation but keep hitting a brick wall! If $X$ and $Y$ are independent both Binomial$(n,\theta)$, how do I work out the conditional expectation of $X$ given that $X+Y=m$ ?
I've tried…
Sam
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1 answer
What is the unconditional distribution?
Suppose that Θ is a random variable that follows a gamma distribution with parameters λ and α, where α is an integer, and suppose that, conditional on Θ, X follows a Poisson distribution with parameter Θ. Find the unconditional distribution of α + X…
Sung Pang
- 27
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0 answers
Conditional Expectations Problem
Let $X$ and $Y$ be r.v. such that $E[X^2] < \infty$, $E[Y^2] < \infty$. Suppose that $E[X|Y] = Y$ and $E[Y|X] = X$. Prove that $Y=X$ with probability 1.
My professor gave us a hint: Compute $E[X-Y]^2$ using conditioning on X and Y.
My attempt:…
Kerry
- 795
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2 answers
Expectation to draw a random variable based upon another random variable
Any help on following question will be much appreciated:
Mr A selects a number X randomly from the uniform distribution on [0,1]. Then Mr B repeatedly and independently draws numbers Y1,Y2 ..... from the uniform distribution on [0,1], until he gets…
Amit K Anand
- 41
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1 answer
Why $\mathbb E[A_n\mid X_1,...,X_{n}]=A_n.$
Let $(A_{n})$ with $n\geq 1$ an non decreasing sequence of random variable such that $A_{n+1}$ dépend of $X_1,...,X_n$ for all $n$. I agree that $$\mathbb E[A_{n+1}\mid X_1,...,X_n]=A_{n+1}$$
But I don't understand why
$$\mathbb E[A_n\mid…
idm
- 11,824
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1 answer
Conditional pmf and mean for geometric rv
I have a geometric r.v. p(1-p)^(n-1), n = 1, 2, 3, 4, ...
I was able to figure out the conditional mean conditioned on X > a. My answer to this is E[X] + a. I am pretty sure that is correct but if not can anyone let me know where I went wrong.
I am…
user100503
- 449
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1 answer
The Notation of Conditional Expectation
Wikipedia told me the fomula for $E(X|Y=y)$, where $y$ is an particular number.
But in my homework, the question is to find $E(X|Y)$, and $Y$ has two values, $1$ and $-1$, under different conditions.
Is it equal to $E(X|Y=1)+E(X|Y=-1)$?
Neuer
- 49
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1 answer
partial derivative of conditional expectation
Can someone help me where to find the necessary background material or explain me why
$\frac{\partial E(X|Y)}{\partial Y} = \frac{\mathrm{Cov}(X,Y)}{\mathrm{Var}(X)}$ for the linear Gaussian case?
