All Questions
Tagged with probability-theory conditional-expectation
1,937
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Expectation of a quotient of random variables
Let $Y$ follow the distribution described by the PDFs $f_Y(y)=2y$ on $(0,1)$. Conditionally on $Y=y$, $X$ follows a uniform distribution on $(0,y)$. Compute $E(X)$ and $E(X/Y)$.
I have calculated the ...
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1
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146
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Question about orthogonality principle
Orthogonality principle states the following:
let $V$ and $W$ be any random variables then for any function $f(W)$
\begin{align}
E[(V-E[V|W])f(W)]=0
\end{align}
My question is wether the following ...
- 6,015
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Can this random variable enter the conditional expectation? Why?
Let $\mathcal{F}_1\subseteq\mathcal{F}_2\subseteq\mathcal{F}_3$ be a filtration, and X, Y two random variables, where X and Y are both $\mathcal{F}_3$-measurable.
Can the expression:
$$\mathbb{E}[(X\...
- 93
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1
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47
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Iteration of conditional expectation conditioned on independent random variables
This comes from Aad Van der Vaart - Asymptotic statistics, Lemma 11.11 (Hoeffding Decomposition)
Suppose I have independent random variables $X_1,...,X_n$. For any random variables $T$ such that $\...
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Find probability mass function and conditional expectation [closed]
Let $X_0,X_1,X_2,...$ be independent identically distributed
nonnegative random variables having a continuous distribution. Let $N$
be the first index $k$ for which $X_k>X_0$. That is, $N=1$ if ...
- 133
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1
answer
817
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Conditional expectation of the product of two dependent random variables
Let $A$, $B$ and $C$ be random variables, with $A$ and $B$ dependent. Is it true that
$\mathbb{E}(A\cdot B|C)=\mathbb{E}(A|C)\cdot\mathbb{E}(B|A,C)$?
In particular, can I say that \begin{equation*}\...
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1
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47
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Distribution of independent random variables [closed]
I have problem with the following:
We have a random variable $X_1$, which is binomially distributed, i.e. $X_1$ ~ $B_{n,p}$. Furthermore $X_2$ is independent from $X_1$ and is binomially distributed ...
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2
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63
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Conditional expectation calculation
Is following equality true:
$$E[X_{1}X_{2}|X_{1}=2]=E[2X_{2}|X_{1}=2]$$
If no how to calculate/simplify this, assuming that we have joint density and marginal densities of those RV and they are not ...
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1
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Prove or disprove that if $X$ is independent of $Y$, then $E[X|F]$ is independent of $E[Y|F]$ [closed]
If $X$ is independent of $Y$, then $E[X|F]$ is independent of $E[Y|F]$.
If the above statement is not correct, how to construct a counter example to disprove it?
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Conditional Expectation Counterexample [closed]
I am trying to think of a counterexample to the following:
If $E(Y|X) =0$, and $E(Y^2|X) = \sigma ^2$, a constant, then $X$ and $Y$ are independent.
Thanks for any help in advance.
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For the random variables X,Y and Z, Is the expression E[E[X|Z]|Y] = E[X|Z] true or False given we do not know if X,Y,Z are independent or not.
Given, we do not know if $X,Y,Z$ are independent or not. Will the expression $E[E[X|Z]|Y] = E[X|Z]$ hold true?
What I have tried:
Now, $E[X|Z]$ is a random variable so let $E[X|Z]$ be $A$
So, problem ...
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1
answer
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Law of iterated expectations applied to a ratio
Consider the random variables $Y, Z_1, Z\equiv(Z_1,\dots,Z_n)$ with $Z_1,...,Z_n$ i.i.d.
Is it true that
$$\mathbb E\left(\frac{Y}{Z_1}\right)=\mathbb E\left(\frac{\mathbb E(Y\mid Z)}{Z_1} \right)$$?...
- 266
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2
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$\mathbb E(X)=\sum \mathbb E(X\mid Y)\mathbb P(Y=n)$ ¿why?
Let $\langle X, Y\rangle$ be a random vector such that $X$ has finite expected value and $Y$ is discrete with values $0.1, \cdots $ such that $\mathbb P (Y = n)> 0$ for $n = 0.1, \cdots$ Show ...
user776949
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1
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Prove that E(E(Y|X)*X)=E(Y*X) [closed]
How can one prove that:
$$ E( E(Y \mid X) \cdot X)=E(Y \cdot X)$$ if $E(Y \mid X)$ is well-defined.
Are we free to use the law of iterated expectations? I am confused since now the expectation of ...
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2
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When does the conditional expectation of the sum of random variables match with the sum their respective conditional expectations?
I am studying stochastic processes. While studying random walk I acquainted with a notation $N_i$ where $$N_i = \mathrm {Total\ number\ of\ times\ of\ visit\ to\ i}.$$ Let $(X_n)_{n \geq 0}$ be a ...
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